Calculus
Textbooks
Boundless Calculus
Differential Equations, Parametric Equations, and Sequences and Series
Calculus Textbooks Boundless Calculus Differential Equations, Parametric Equations, and Sequences and Series
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus

Section 3

Infinite Sequences and Series

Book Version 1
By Boundless
Boundless Calculus
Calculus
by Boundless
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13 concepts
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Sequences

A sequence is an ordered list of objects and can be considered as a function whose domain is the natural numbers.

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Series

A series is the sum of the terms of a sequence.

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The Integral Test and Estimates of Sums

The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.

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Comparison Tests

Comparison test may mean either limit comparison test or direct comparison test, both of which can be used to test convergence of a series.

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Alternating Series

An alternating series is an infinite series of the form $\sum_{n=0}^\infty (-1)^n\,a_n$ or $\sum_{n=0}^\infty (-1)^{n-1}\,a_n$ with $a_n > 0$ for all $n$.

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Absolute Convergence and Ratio and Root Tests

An infinite series of numbers is said to converge absolutely if the sum of the absolute value of the summand is finite.

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Tips for Testing Series

Convergence tests are methods of testing for the convergence or divergence of an infinite series.

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Power Series

A power series (in one variable) is an infinite series of the form $f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n$, where $a_n$ is the coefficient of the $n$th term and $x$ varies around $c$.

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Expressing Functions as Power Functions

A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.

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Taylor and Maclaurin Series

Taylor series represents a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point.

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Applications of Taylor Series

Taylor series expansion can help approximating values of functions and evaluating definite integrals.

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Summing an Infinite Series

Infinite sequences and series can either converge or diverge.

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Convergence of Series with Positive Terms

For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sum $\sum_{n=0}^{\infty}a_n$ can either converge or diverge to $\infty$.

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