AP Calculus BC - Unit 10: Infinite Sequences and Series

Convergence Tests for Series:

Test Name	The Series ...	Will Converge If	Or Will Diverge If	Comments
$n^{\text{th}}$ -term test	$\sum a_n$	-	$\lim_{n \to \infty} a_n \neq 0$	For divergence only; the converse is false.
Geometric	$\sum ar^{n-1}$	$\|r\| < 1$	$\|r\| \geq 1$	Sum = $\frac{a}{1 - r}$
Alternating Series	$\sum (-1)^{n-1} a_n$	$\|a_{n+1}\| < \|a_n\|$ and $\lim_{n \to \infty} a_n = 0$	-	Error bound: $\|S - S_n\| \leq \|a_{n+1}\|$
Integral Test	$\sum a_n$ , where $a_n = f(n) \geq 0$	$\int_1^\infty f(x) dx$ converges	$\int_1^\infty f(x) dx$ diverges	$f$ must be continuous, positive, and decreasing.
p-Series	$\sum \frac{1}{n^p}$	$p > 1$	$p \leq 1$	-

Test Name	The Series ...	Will Converge If	Or Will Diverge If	Comments
Limit Comparison	$\sum a_n$	$\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0$ and $\sum b_n$ converges	$\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0$ and $\sum b_n$ diverges	-
Root Test	$\sum a_n$	$\lim_{n \to \infty} \sqrt[n]{a_n} < 1$	$\lim_{n \to \infty} \sqrt[n]{a_n} > 1$	Cannot be used if $\lim_{n \to \infty} \sqrt[n]{a_n} = 1$

The Maclaurin Series for common functions:

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$

To find the interval of convergence of a power series, apply the **Ratio Test** and check endpoints separately.

The Lagrange Remainder for the $n^{\text{th}}$ term of a Taylor Series is:

$R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!} (x - c)^{n+1}$

where $z$ is a value between $x$ and $c$ .