Definition of a Function

Definition of a function. Evaluation of a function. Representing a function, e.g., verbally, numerical, algebraically, graphically, piecewise, recursive, parametric, integral, and series representation. The Vertical Line Test. Read more... 2806 words,🕔14 minutes read, May 14, 2022.

Graphing functions

A comprehensive guide to graphing mathematical functions covering systematic procedures from domain analysis to plotting key features. Detailed coverage of linear, quadratic, rational, exponential, logarithmic, trigonometric, and piecewise functions with emphasis on intercepts, asymptotes, symmetry, and continuity. Read more... 1862 words,🕔9 minutes read, May 14, 2022.

Intercepts of Graphs

A detailed guide to finding and interpreting x-intercepts and y-intercepts. Covers linear, quadratic, exponential, and trigonometric functions, as well as circles. Explains how to find intercepts from equations and graphs, and how to use them for graphing lines. Read more... 2033 words,🕔10 minutes read, May 14, 2022.

Monotonic Functions and Local Extrema

A comprehensive guide to monotonic functions (increasing, decreasing, constant) and their relationship with derivatives. Explores strict monotonicity conditions, critical points, and local extrema with practical methods for finding maxima and minima. Includes detailed examples like e^x, sqrt(x), absolute value, and piecewise functions. Read more... 2315 words,🕔11 minutes read, May 14, 2022.

Local Extrema. First and Second Derivative Tests

A detailed guide to finding local maxima and minima of functions. Covers the First Derivative Test, including its proof using the Mean Value Theorem, and the Second Derivative Test for classifying critical points. Includes worked examples with polynomial and trigonometric functions. Read more... 2101 words,🕔10 minutes read, May 14, 2022.

Asymptotes and End Behavior

A detailed guide to understanding asymptotes and the end behavior of functions. Covers vertical, horizontal, and oblique asymptotes, with a focus on rational, exponential, and logarithmic functions. Includes solved examples and graphical analysis. Read more... 3264 words,🕔16 minutes read, Jul 14, 2022.

Analyzing and Graphing Rational Functions

Provides detailed, step-by-step examples for analyzing and graphing rational functions. Covers finding domain, range, intercepts, asymptotes (including oblique), and using first and second derivative tests to determine monotonicity and concavity for accurate sketching Read more... 2568 words,🕔13 minutes read, Jul 14, 2022.

Increasing and Decreasing Functions

A comprehensive guide to understanding monotonic functions. Covers the intuitive and formal definitions of increasing and decreasing functions, explains the derivative test for monotonicity, and details how to find increasing and decreasing intervals by analyzing critical points and the sign of the derivative. Includes solved examples with linear, quadratic, cubic, exponential, and rational functions. Read more... 3854 words,🕔19 minutes read, Jul 14, 2022.

Algebra of functions

Comprehensive guide to algebra of functions covering arithmetic operations (sum, difference, product, quotient), composition, differentiation rules, limits, horizontal and vertical asymptotes, and the test-point method for sign analysis with detailed examples. Read more... 2988 words,🕔15 minutes read, May 14, 2022.

Analyzing Concavity with the Second Derivative Test

A comprehensive guide to understanding concavity and inflection points. Explains the geometric and analytic definitions, connects them to the second derivative, and provides detailed examples for quadratic, cubic, rational, trigonometric, and logarithmic functions to illustrate how to determine intervals of concavity and locate inflection points. Read more... 2862 words,🕔14 minutes read, May 14, 2022.

Exponential Functions. Growth, Decay, and Properties

Comprehensive exploration of exponential functions including definition, properties, derivatives, Taylor series, and applications. Covers exponential growth vs decay, the natural exponential e^x, and techniques for solving exponential equations. Read more... 2880 words,🕔14 minutes read, May 14, 2022.

Limits in Calculus

Comprehensive introduction to limits covering intuitive and formal epsilon-delta definitions, one-sided limits, evaluating limits from graphs, and techniques for solving limits including factoring and rationalizing. Read more... 3046 words,🕔15 minutes read, May 14, 2022.

Epsilon-Delta Proofs for Limit

Comprehensive collection of solved limit problems using rigorous epsilon-delta proofs. Detailed step-by-step solutions for polynomial, rational, radical, and trigonometric limits with careful epsilon and delta selection strategies. Read more... 1107 words,🕔6 minutes read, May 14, 2022.

Inverse functions

Definition, graphical interpretation, and the horizontal line test. Learn how to find inverses of linear, rational, exponential, and trigonometric functions. Calculate and understand the derivative of inverse functions with solved examples. Read more... 3027 words,🕔15 minutes read, May 14, 2022.

Logarithm functions

Explores the logarithm function as the inverse of exponentiation. Covers logarithmic properties (product, quotient, power), domain, range, vertical asymptotes, and monotonicity. Includes derivatives, integrals, Taylor series expansion, and solved examples of logarithmic equations. Read more... 3330 words,🕔16 minutes read, May 14, 2022.

Symmetry of Graphs

Comprehensive guide to symmetry of graphs covering even and odd functions, symmetry about axes and origin, vertical line symmetry, point symmetry, and algebraic tests. Includes even-odd decomposition formula and solved examples for various function types. Read more... 3471 words,🕔17 minutes read, May 14, 2022.

Types of functions

Comprehensive classification of mathematical functions covering linear, quadratic, polynomial, exponential, logarithmic, trigonometric, rational, piecewise, hyperbolic, and special functions. Discusses classification criteria including mapping, symmetry, smoothness, growth rates, monotonicity, and explores implicit, parametric, and multivariable functions. Read more... 4545 words,🕔22 minutes read, May 14, 2022.

Absolute Extrema

Definitions, the Extreme Value Theorem with proof, and solved examples for finding absolute extrema of functions. Read more... 2888 words,🕔14 minutes read, Apr 14, 2022.

Continuity and discontinuity

Definitions of continuity and discontinuity, including ε–δ formulation, types of discontinuities (jump, removable, infinite, mixed), examples with various functions, and proof that differentiability implies continuity. Read more... 2374 words,🕔12 minutes read, May 14, 2022.

Infinite Limits

Intuitive and formal definitions of infinite limits using epsilon-delta, vertical asymptotes, one-sided infinite limits, and solved examples with proofs. Read more... 1493 words,🕔8 minutes read, May 14, 2022.

Limits at infinity

Intuitive and formal definitions of limits at infinity with epsilon-delta proofs, horizontal asymptotes, growth rate hierarchies, rational functions using dominant term method, limits with radicals, L'Hôpital's Rule, and extensive solved examples. Read more... 1708 words,🕔9 minutes read, May 14, 2022.

Limits of Rational Functions and at Infinity

Definition of rational functions, techniques for computing limits including direct substitution, factoring, rationalization, and limits at infinity using the dominant term method with extensive solved examples. Read more... 2022 words,🕔10 minutes read, May 14, 2022.

One-Sided Limits

Intuitive and formal definitions of one-sided limits, relationship with two-sided limits, extensive solved examples with piecewise functions, absolute value, floor function, and epsilon-delta proofs. Read more... 2529 words,🕔12 minutes read, May 14, 2022.

Strategy in finding limits

Comprehensive strategies for calculating limits including L'Hôpital's Rule for various indeterminate forms, the Squeeze Theorem with geometric proofs, and techniques for trigonometric limits using identities and fundamental limits. Read more... 2309 words,🕔11 minutes read, May 14, 2022.

The Squeeze Theorem

The Squeeze Theorem with intuitive explanation, formal definition, proof, and extensive solved examples including oscillatory functions, trigonometric limits, and limits at infinity. Read more... 1495 words,🕔8 minutes read, May 14, 2022.

Derivate and Tangent Lines

Definition of derivative via secant lines, geometric meaning (tangent line), solved exercises for functions like 1/x, √x, |x|, xⁿ, tangent line equations, differentials, and applications in physics and population dynamics. Read more... 2525 words,🕔12 minutes read, May 14, 2022.

The Limit Laws

Definitions, proofs, and examples of fundamental limit laws including constant, sum, difference, product, quotient, power, root, polynomial, and composite function limits. Read more... 2187 words,🕔11 minutes read, May 14, 2022.

Derivate of Exponential and Logarithms

Derivative of exponential and logarithmic functions. Definition, proofs, and solved examples including a^x, e^x, ln(x), log_a(x), and power rule for real exponents. Read more... 1306 words,🕔7 minutes read, May 14, 2022.

General Differentiation rules

Derivation and proofs of fundamental differentiation rules including constant, constant multiple, power, sum, product, quotient, and chain rules with examples. Read more... 1593 words,🕔8 minutes read, May 14, 2022.

Derivate Trigonometric Functions

Basic trigonometry including right triangle definitions, unit circle, degrees and radians, special angles, common trigonometric identities, and derivatives of trigonometric functions. Read more... 2201 words,🕔11 minutes read, May 14, 2022.

Derivate of inverse functions

Derivatives of inverse functions using the Inverse Function Theorem, including inverse trigonometric functions (arctan, arcsin, arccos, arcsec) with proofs and solved examples. Read more... 2367 words,🕔12 minutes read, May 14, 2022.

Higher derivatives

Higher Order Derivatives. Definition, interpretation, notations, applications (Taylor series), and solved exercises. Read more... 2142 words,🕔11 minutes read, May 14, 2022.

Implicit Differentiation

Implicit Differentiation. Definition, techniques (chain rule, logarithmic differentiation), inverse functions, and solved exercises for finding derivatives without explicit functions. Read more... 2042 words,🕔10 minutes read, May 14, 2022.

Linear and Quadratic Approximation

Comprehensive guide to linear and quadratic approximations. Covers the tangent line formula, quadratic approximation, essential approximations (sin x ≈ x, e^x ≈ 1+x, etc.), error analysis, applications to limits, and connection to Taylor series with remainder bounds. Includes step-by-step examples and comparative error tables. Read more... 2883 words,🕔14 minutes read, May 14, 2022.

Integration of Trigonometric Functions II

Integration of Trigonometric Functions. Formulas and solved examples. Trigonometry substitution for integrals. Completing the square. Read more... 814 words,🕔4 minutes read, May 14, 2022.

Properties of Integrals and Change of Variables

A detailed guide to the fundamental properties of definite and indefinite integrals. Covers linearity, interval reversal, adjacent intervals, and the powerful symmetry rules for even and odd functions. Includes the Comparison Theorem, step-by-step integration by substitution (including trigonometric substitution for the area of a circle), and a critical look at the pitfall of integration by substitution (changing signs in u'). Read more... 1780 words,🕔9 minutes read, May 14, 2022.

The Fundamental Theorems of Calculus and the MVT for Integrals

A rigorous exploration of the Fundamental Theorems of Calculus. Covers the Net Change Theorem, the Mean Value Theorem for Integrals (with proof), and the Second FTC establishing the derivative of an integral. Concludes with the Evaluation Theorem and solved exercises applying these concepts to accumulation functions and the Chain Rule. Read more... 1881 words,🕔9 minutes read, May 14, 2022.

Functions and Graphing Fundamentals

Fundamentals of graphing functions, including linear equations, intercepts, and quadratic functions (parabolas, vertex, discriminant). Progresses to advanced curve sketching using calculus, analyzing critical points, concavity, and identifying vertical, horizontal, and oblique asymptotes in rational and cubic functions. Read more... 1855 words,🕔9 minutes read, May 14, 2022.

Understanding Functions and Their Graphs

A comprehensive guide to understanding and graphing functions. Covers a 7-step calculus strategy for curve sketching, basic function transformations (shifts, stretches, reflections), graphing piecewise functions with discontinuities, finding and plotting inverse functions, and analyzing exotic functions like the Topologist's sine curve using the Squeeze Theorem. Read more... 2591 words,🕔13 minutes read, May 14, 2022.

Sequences, Subsequences, and the Bolzano-Weierstrass Theorem

An in-depth exploration of sequences and their limits, featuring a proof of the Bolzano-Weierstrass theorem via the bisection method and nested intervals. Includes an advanced application proving that the chaotic sequence sin(n) contains a convergent subsequence using the continued fraction algorithm and Diophantine approximation. Read more... 2065 words,🕔10 minutes read, May 14, 2022.

The Boundedness Theorem

A rigorous exploration of the Boundedness Theorem. Begins with the definition of differentiability, proves that limits of sequences in closed intervals remain within those intervals, and uses the Bolzano-Weierstrass theorem to formally prove the Boundedness Theorem. Concludes with counterexamples showing why continuity and closed, bounded domains are strict requirements. Read more... 1231 words,🕔6 minutes read, May 14, 2022.

Derivatives as Rates of Change

A comprehensive guide to derivatives as rates of change. Covers the formal definition of differentiability, essential differentiation rules, critical points, and linear approximation. Features a step-by-step methodology and solved related rates problems involving kinematics, expanding geometric shapes, and similar triangles. Read more... 2381 words,🕔12 minutes read, May 14, 2022.

Derivatives as Rates of Change II

Advanced applications of derivatives as rates of change. Features detailed solved exercises involving trigonometric tracking (flashlights, rockets, and police radar), 3D geometric constraints (melting ice cubes and filling conical tanks), unit conversions, and population estimation using linear approximation. Read more... 2202 words,🕔11 minutes read, May 14, 2022.

Solving Related Rates Problems

A comprehensive guide to solving related rates problems in calculus. Reviews the definition of derivatives as instantaneous rates of change, outlines a universal 4-step methodology, and applies it to 8 detailed solved exercises including the leaking sand cone, sliding ladder, approaching cars, expanding sphere, police chase, rocket launch, filling a reservoir, and the airplane observer problem. Read more... 2373 words,🕔12 minutes read, May 14, 2022.

Fermat's Theorem and Absolute Extrema

An in-depth look at Fermat's Theorem, its proof, and why the converse is not always true. Details the 4-step process for finding absolute extrema on a closed interval using the Extreme Value Theorem, complete with solved exercises covering polynomials, absolute values, and piecewise functions. Read more... 1875 words,🕔9 minutes read, May 14, 2022.

Newton's Method (The Newton-Raphson Method)

A comprehensive guide to Newton's Method (Newton-Raphson). Covers the geometric derivation of the iterative formula using tangent lines, discusses its limitations, and features 5 step-by-step solved examples including approximating square/cube roots, finding polynomial roots, and calculating curve intersections. Read more... 1589 words,🕔8 minutes read, May 14, 2022.

Optimization Problems in Calculus

A comprehensive guide to solving optimization problems using calculus. Features step-by-step solved exercises including minimizing the sum of numbers with a fixed product, optimizing areas of circles and squares cut from a single wire, maximizing rectangular areas, and applying the Extreme Value Theorem to a tricky two-square wire problem. Read more... 2103 words,🕔10 minutes read, May 14, 2022.

Advanced Optimization Problems in Calculus

Advanced applications of derivatives in geometric optimization. Features 7 step-by-step solved exercises including maximizing area against a rock wall, minimizing surface area of open-top and folded boxes, optimizing bordered gardens, maximizing a rectangle inscribed in an ellipse using parametric equations, and optimizing areas when cutting wire into a square and an equilateral triangle. Read more... 2531 words,🕔12 minutes read, May 14, 2022.

Fundamental Theorems of Calculus (EVT, IVT, Rolle's, and MVT)

A comprehensive guide to the foundational theorems of differential calculus. Covers the Extreme Value Theorem, Intermediate Value Theorem for root finding, Rolle's Theorem, and the Mean Value Theorem with geometric interpretations and formal proofs. Also explores monotonicity (increasing and decreasing functions) and uses derivatives to prove exponential inequalities connected to Taylor/Maclaurin series. Read more... 2386 words,🕔12 minutes read, May 14, 2022.

The Extreme Value Theorem. Proof, Foundations, & Exercises.

A rigorous exploration of the Extreme Value Theorem. Covers the foundational concepts of upper bounds, the Axiom of Completeness, and the Nested Interval Property. Includes the full formal proof of the EVT using the Bolzano-Weierstrass theorem, followed by the closed-interval method for finding absolute extrema with solved exercises. Read more... 1864 words,🕔9 minutes read, May 14, 2022.

Antiderivatives and Basic Integration Technique

A comprehensive introduction to antiderivatives and indefinite integrals. Covers the constant of integration, fundamental integration rules, algebraic simplification before integrating, integration by substitution (u-substitution), integration by parts, and concludes with a proof of the uniqueness of antiderivatives. Read more... 1550 words,🕔8 minutes read, May 14, 2022.

Definite Integrals and the Fundamental Theorem of Calculus

A comprehensive guide to definite integrals and the Fundamental Theorem of Calculus (FTC). Covers the geometric motivation behind accumulation, Riemann sum approximations, an intuitive driving analogy for the FTC, the crucial difference between net signed area and total geometric area, and step-by-step examples for finding areas bounded by curves. Read more... 2088 words,🕔10 minutes read, May 14, 2022.

Evaluating Definite Integrals: Area Problems and U-Substitution

A practical exercise guide to evaluating definite integrals. Features step-by-step examples for direct integration of polynomials and trigonometric functions, calculating total geometric area versus net signed area, finding areas bounded by curves and axes, and applying u-substitution with changed limits for exponential and radical integrands. Read more... 1246 words,🕔6 minutes read, May 14, 2022.

The Logarithm Defined as an Integral

A rigorous calculus-based approach to defining the natural logarithm as a definite integral. Features proofs of the product and power rules of logarithms using integration by substitution. Also introduces the error function as a solution to a non-elementary integral, analyzing its monotonicity, symmetry, and defining the standard erf(x) function along with its sigmoid shape and horizontal asymptotes. Read more... 1670 words,🕔8 minutes read, May 14, 2022.

Finding Areas Between Curves. Vertical and Horizontal Slices

A comprehensive guide to calculating the area between two curves. Details the standard vertical slice method, the advantage of horizontal slices (integrating with respect to y), how to handle regions requiring split integrals at interior intersection points, and using geometric symmetry to drastically simplify complex bounded regions. Read more... 2272 words,🕔11 minutes read, May 14, 2022.

Mean Value Theorem for Integrals

Learn how to calculate the average value of a continuous function over an interval using definite integrals. Includes the derivation, the Mean Value Theorem for Integrals, and step-by-step solved exercises. Read more... 1850 words,🕔9 minutes read, May 14, 2022.

Determining volumes

Learn how to calculate the volume of solids using integration. Covers the disk, washer, and shell methods for solids of revolution, plus step-by-step derivations for the volumes of spheres, cylinders, and pyramids. Read more... 1890 words,🕔9 minutes read, May 14, 2022.

Calculating volumes II

Learn how to calculate the volume of complex solids using integration. This guide covers the General Slicing Method with various cross-sections, as well as the Disk, Washer, and Shell methods. Features step-by-step solved exercises, including volumes requiring inverse trigonometric integration and polynomial expansion. Read more... 2243 words,🕔11 minutes read, May 14, 2022.

Solved Exercises. Determining Volumes

Practice calculating the volume of solids with these step-by-step solved exercises. Covers the cylindrical shell method, disk and washer methods (including logarithmic bounds and horizontal slicing with respect to y), and the general slicing method with known cross-sections like squares and isosceles right triangles. Read more... 1606 words,🕔8 minutes read, May 14, 2022.

Numerical Integration and the Gaussian Integral

An introduction to numerical integration starting with the famous Gaussian Integral. Learn how to approximate definite integrals when elementary antiderivatives do not exist, using Left and Right Riemann Sums, the Trapezoidal Rule, and Simpson's Rule, complete with step-by-step solved examples. Read more... 1936 words,🕔10 minutes read, May 14, 2022.

The Average Value of a Function

Learn how to calculate the average value of a continuous function using definite integrals. This guide progresses from the Average Value Theorem to discrete weighted averages, and finally to continuous weighted averages, featuring step-by-step solved exercises calculating the average temperature of fluids inside parabolic and conical cauldrons. Read more... 1906 words,🕔9 minutes read, May 14, 2022.

Weighted Average II

Explore advanced applications of continuous weighted averages and probability density functions in calculus. Features solved exercises calculating the weighted average temperature in a cylindrical layer, and determining continuous probabilities using parabolic densities and 2D normal distributions (radial probability, golf ball targeting, and the classic MIT dartboard problem). Read more... 1825 words,🕔9 minutes read, May 14, 2022.

Trigonometric Substitution and Integration

Master the integration of trigonometric functions with this comprehensive calculus guide. Covers essential trig identities, the step-by-step trigonometric substitution method for radical expressions, completing the square, and specific strategies for integrating odd and even powers of sine and cosine. Read more... 1678 words,🕔8 minutes read, May 14, 2022.

Arc Length and Surface Area of Revolution

Master the calculus of curves with this complete guide to Arc Length and Surface Area of Revolution. Covers the arc length formula, deriving surface area for both x-axis and y-axis rotations, and features advanced solved exercises including the rigorous integration by parts for sec^3(x) and sec^5(x), concluding with the geometric proof for the surface area of a sphere. Read more... 2064 words,🕔10 minutes read, May 14, 2022.

Integration by Parts

Master essential integration techniques with this comprehensive guide. Covers U-substitution, the Integration by Parts formula, the LIATE mnemonic, a beautiful geometric interpretation of the formula, and advanced solved exercises including repeated integration and deriving reduction formulas for logarithmic and exponential functions. Read more... 1730 words,🕔9 minutes read, May 14, 2022.

Integration by parts II

Master advanced applications of integration by parts with these step-by-step solved exercises. Covers cyclic integrals, integrating inverse trigonometric functions (arcsin, arctan), definite integrals with logarithms, calculating the volume of an exponential wineglass using both shells and disks, and concludes with a rigorous proof of the Gamma function using mathematical induction. Read more... 1880 words,🕔9 minutes read, May 14, 2022.

Integration of Rational Functions

Master the integration of rational functions using partial fraction decomposition. This guide provides step-by-step solved exercises covering the cover-up method for distinct linear factors, handling repeated linear factors and irreducible quadratics, and simplifying improper rational fractions using polynomial long division. Read more... 1330 words,🕔7 minutes read, May 14, 2022.

Parametric Curves and Polar Coordinates

Master parametric equations and polar coordinates in calculus. Learn how to define circles and ellipses parametrically, calculate arc length and surface area, convert between Cartesian and polar systems, graph complex shapes like rose curves and spirals, and derive the formula to find the area enclosed by polar curves. Read more... 1770 words,🕔9 minutes read, May 14, 2022.

L'Hôpital's Rule and Indeterminate Forms

Master the evaluation of tricky calculus limits. This guide starts with the intuition behind limits and basic strategies like direct substitution and factoring. It then dives deep into L'Hôpital's Rule, explaining its derivation, applying it to standard and tricky indeterminate forms (0·∞, ∞-∞), exploring Taylor series alternatives, and highlighting the common pitfalls where L'Hôpital's Rule illegally fails. Read more... 1545 words,🕔8 minutes read, May 14, 2022.

L'Hôpital's Rule II

Explore advanced applications of L'Hôpital's Rule to evaluate complex indeterminate forms including $1^\infty$, $0^0$, and $0 \cdot \infty$. Then, discover how calculus limits directly connect to computer science through Asymptotic Complexity, analyzing the hierarchy of growth, and understanding Big O, Omega, and Theta notations for algorithm efficiency. Read more... 1945 words,🕔10 minutes read, May 14, 2022.

Comparison Test For Improper Integrals

Learn how to determine the convergence or divergence of improper integrals without explicitly evaluating them. This guide covers core convergence theorems (p-series, exponential decay), the Direct Comparison Test for bounding functions, provides a formal proof of the Limit Comparison Test, and features step-by-step solved examples including the Gaussian integral and trigonometric bounds. Read more... 1800 words,🕔9 minutes read, May 14, 2022.

Definite and Improper Integrals

Master definite and improper integrals in calculus. This guide defines standard integration, explains when integrals become improper due to infinite bounds or vertical asymptotes, and distinguishes between convergence and divergence. Features step-by-step solved exercises using U-substitution, integration by parts, L'Hôpital's rule, and the General p-Test. Read more... 1909 words,🕔9 minutes read, May 14, 2022.

Improper Integrals. Type II (Vertical Asymptotes)

Learn how to evaluate Type II improper integrals where integrands approach infinity due to vertical asymptotes. This guide covers the formal definitions for endpoint and interior discontinuities, the hierarchy of asymptotes near zero, and features step-by-step solved exercises including mixed Type I & II integrals, partial fractions, and integration by parts with L'Hôpital's rule. Read more... 2121 words,🕔10 minutes read, May 14, 2022.

Alternating Series

Master the convergence of alternating series in calculus. This guide clearly defines absolute versus conditional convergence, provides a rigorous proof of the Alternating Series Test using the Monotone Convergence Theorem, and features step-by-step solved exercises demonstrating eventual monotonicity and how to determine if an alternating series converges conditionally or absolutely. Read more... 1925 words,🕔10 minutes read, May 14, 2022.

Sequences and Infinite Series

Master the fundamentals of sequences and infinite series in calculus. This guide defines convergence and divergence through partial sums, provides formal derivations for Arithmetic and Geometric series formulas, explains how to collapse Telescoping series using partial fractions, and covers the algebraic properties of convergent series. Read more... 1834 words,🕔9 minutes read, May 14, 2022.

Testing for Convergence: The Divergence and Integral Tests

Learn how to determine if an infinite series converges or diverges using two foundational calculus tests. This guide covers the Necessary Condition for Convergence (the Divergence Test), provides a rigorous proof using Riemann sums for the Harmonic series, details the Integral Test and its error bounds, and features step-by-step solved exercises evaluating logarithmic, exponential, and arctangent series. Read more... 2194 words,🕔11 minutes read, May 14, 2022.

Comparison Tests for Series Convergence

Master the Direct and Limit Comparison Tests for infinite series. This guide features rigorous proofs, including the Monotone Convergence Theorem and the p-series convergence proof via the Integral Test. It includes step-by-step solved exercises for establishing series convergence, and concludes with a fascinating real-world application of the divergent harmonic series, calculating the theoretical limit of overhanging blocks. Read more... 2204 words,🕔11 minutes read, May 14, 2022.

Power Series. Convergence, derivatives, and Integrals

Master power series in calculus. This guide rigorously proves the three cases of power series convergence using the Least Upper Bound Property, demonstrates how to find intervals of convergence using the Ratio and Root tests, shows how to construct power representations from rational functions, and explains how to differentiate and integrate power series term-by-term. Read more... 2459 words,🕔12 minutes read, May 14, 2022.

The Ratio and Root Tests for Series Convergence

Master the Ratio and Root Tests for determining the convergence of infinite series. This guide provides formal definitions, rigorous proofs for both tests, and features step-by-step solved examples demonstrating how to find the radius and interval of convergence for series involving factorials, shifted power series, and complex limits. Read more... 1737 words,🕔9 minutes read, May 14, 2022.

Taylor’s Formula and Series

Approximating functions with polynomials using Taylor's Formula. Covers Taylor and Maclaurin series, Peano and Lagrange remainders, and includes solved exercises for common functions. Read more... 2083 words,🕔10 minutes read, May 14, 2022.