AP®MathsCollege BoardCalculus BCStudy GuidesUnit 10: Infinite Sequences and SeriesUnit 10 OverviewUnit 10 Summary

Unit 10 Summary (College Board AP® Calculus BC): Study Guide

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 20 May 2026

Infinite sequences and series summary

Key definitions

A sequence is an ordered collection of numbers denoted $\{a_{n}\}$
A partial sum $s_{n}$ is the sum of the first $n$ terms of a sequence
A series $\sum_{n = 0}^{\infty} a_{n}$ is the sum of terms in a sequence
An infinite series $\sum_{n = 0}^{n} a_{n}$ converges if $\lim_{n \to \infty} s_{n} = \lim_{n \to \infty} \sum_{k = 0}^{n} a_{k}$ exists
An infinite series diverges if it does not converge
A geometric series is of the form $\sum_{n = 0}^{\infty} a \cdot r^{n} = a + a r + a r^{2} + . . .$
- $a \neq 0$ is the first term
- $r$ is the common ratio
The harmonic series is $\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + . . .$
The alternating harmonic series is $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - . . .$
A p-series is $\sum_{n = 1}^{\infty} \frac{1}{n^{p}} = 1 + \frac{1}{2^{p}} + \frac{1}{3^{p}} + . . .$
A series $\sum_{n = 0}^{\infty} a_{n}$ converges absolutely if $\sum_{n = 0}^{\infty} |a_{n}|$ converges
A series $\sum_{n = 0}^{\infty} a_{n}$ converges conditionally if $\sum_{n = 0}^{\infty} a_{n}$ converges but $\sum_{n = 0}^{\infty} |a_{n}|$ diverges
A Taylor polynomial of degree $n$ of a function $f$ about the point $x = a$ is $p_{n} (x) = f (a) + f' (a) \cdot (x - a) + f'' (a) \cdot \frac{{(x - a)}^{2}}{2!} + . . . + f^{(n)} (a) \cdot \frac{{(x - a)}^{n}}{n!}$
- Provided the $n$ derivatives of $f$ exist at $x = a$
A Maclaurin polynomial of degree $n$ of a function $f$ is $p_{n} (x) = f (0) + f' (0) \cdot x + f'' (0) \cdot \frac{x^{2}}{2!} + . . . + f^{(n)} (0) \cdot \frac{x^{n}}{n!}$
- Provided the $n$ derivatives of $f$ exist at $x = 0$
A power series is a series that depends on a variable $\sum_{n = 0}^{\infty} a_{n} \cdot {(x - r)}^{n} = a_{0} + a_{1} \cdot (x - r) + a_{2} \cdot {(x - r)}^{2} + . . .$
The interval of convergence of a power series is the set of values for which the power series converges
The radius of convergence is half the length of the interval of convergence
A Taylor series for a function $f$ about point $x = a$ is $\sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} \cdot {(x - a)}^{n} = f (a) + f' (a) \cdot (x - a) + \frac{f'' (a)}{2!} \cdot {(x - a)}^{2} + . . .$
A Maclaurin series for a function $f$ is $\sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} \cdot x^{n} = f (0) + f' (0) \cdot x + \frac{f'' (0)}{2!} \cdot x^{2} + . . .$

Key tests

nth term test
- If $\lim_{n \to \infty} a_{n} \neq 0$ , then $\sum_{n = 0}^{\infty} a_{n}$ diverges
Integral test
- Given that $f$ is a continuous, positive, decreasing function on $[c, \infty)$ such that $a_{n} = f (n)$
- If $\int_{c}^{\infty} f (x) d x$ exists, then $\sum_{n = c}^{\infty} a_{n}$ converges
- If $\int_{c}^{\infty} f (x) d x$ does not exist, then $\sum_{n = c}^{\infty} a_{n}$ diverges
Comparison test
- Given that $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ are two series with non-negative terms
- If $\sum_{n = 1}^{\infty} b_{n}$ converges and if $a_{n} \leq b_{n}$ for all $n$ , then $\sum_{n = 1}^{\infty} a_{n}$ converges
- If $\sum_{n = 1}^{\infty} b_{n}$ diverges and if $a_{n} \geq b_{n}$ for all $n$ , then $\sum_{n = 1}^{\infty} a_{n}$ diverges
Limit comparison test
- Given that $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ are two series with non-negative terms
- If $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = L$ , where $0 < L < \infty$ , then either both series converge or both series diverge
Ratio test
- $\sum_{n = 1}^{\infty} a_{n}$
- If $\lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| < 1$ , then $\sum_{n = 1}^{\infty} a_{n}$ converges absolutely
- If $\lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| > 1$ or if the limit is infinite, then $\sum_{n = 1}^{\infty} a_{n}$ diverges
- If $\lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| = 1$ , then the ratio test provides no information about convergence
Alternating series test
- Given an alternating series of the form $\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} \cdot a_{n}$ or $\sum_{n = 1}^{\infty} {(- 1)}^{n} \cdot a_{n}$ with $a_{n} > 0$ for each value of $n$
- The series converges if $a_{1} \geq a_{2} \geq a_{3} \geq . . . \geq a_{n} \geq . . .$ and $\lim_{n \to \infty} a_{n} = 0$

Key formulas

If $\sum_{n = 0}^{\infty} a_{n} = A$ and $\sum_{n = 0}^{\infty} b_{n} = B$ , where $A$ and $B$ are finite, then:
- $\sum_{n = 0}^{\infty} (c a_{n}) = c A$ for any real number $c$
- $\sum_{n = 0}^{\infty} (a_{n} + b_{n}) = A + B$
$\sum_{n = 0}^{\infty} a \cdot r^{n} = \frac{a}{1 - r}$ provided $|r| < 1$
The alternating series error bound for $S = \sum_{n = 1}^{\infty} {(- 1)}^{n} \cdot a_{n}$ or $S = \sum_{n = 1}^{\infty} {(- 1)}^{n + 1} \cdot a_{n}$ is given by $|S - s_{n}| \leq a_{n + 1}$ where $s_{n} = \sum_{k = 1}^{n} {(- 1)}^{k} \cdot a_{k}$ or $s_{n} = \sum_{k = 1}^{n} {(- 1)}^{k + 1} \cdot a_{k}$
The Lagrange error bound of a Taylor polynomial of degree $p_{n}$ for a function $f$ about $x = a$ is given by $|f (x) - p_{n} (x)| \leq \frac{M}{(n + 1)!} {|x - a|}^{n + 1}$
- Where $|f^{(n + 1)} (x)| \leq M$ for all $x$ in the interval

Key facts

The following table gives the convergent conditions for common series

Series	Convergent condition
Geometric $\sum_{n = 0}^{\infty} a \cdot r^{n} = a + a r + a r^{2} + . . .$	Absolutely convergent when $\|r\| < 1$ Divergent when $\|r\| \geq 1$
Harmonic $\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + . . .$	Divergent
Alternating harmonic $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - . . .$	Conditionally convergent
p-series $\sum_{n = 1}^{\infty} \frac{1}{n^{p}} = 1 + \frac{1}{2^{p}} + \frac{1}{3^{p}} + . . .$	Absolutely convergent when $p > 1$ Divergent when $p \leq 1$

If a power series converges, then one of the following is true:
- it converges only at a single point
- it converges for all values in an interval
- it converges for all values
The Maclaurin series or a Taylor series of a function can be used to approximate the function at values within its interval of convergence
The table below shows common Maclaurin series with their interval of convergence

Function	Maclaurin or Taylor series	Interval of convergence
$\frac{1}{1 - x}$	$\sum_{n = 0}^{\infty} x^{n} = 1 + x + x^{2} + . . .$	$- 1 < x < 1$
$e^{x}$	$\sum_{n = 0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + . . .$	$- \infty < x < \infty$
$\sin x$	$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n + 1} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - . . .$	$- \infty < x < \infty$
$\cos x$	$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} x^{2 n} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - . . .$	$- \infty < x < \infty$

A power series can be differentiated or integrated term-by-term within its interval of convergence

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