APEX Summary of Tests for Series

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Section B.7 Summary of Tests for Series

Table B.7.1.

Test	Series	Condition(s) of Convergence 🔗	Condition(s) of Divergence 🔗	Comment
$n$ th-Term	$\sum_{n = 1}^{\infty} a_{n}$		$lim_{n \to \infty} a_{n} \neq 0$ 🔗	Cannot be used to show convergence. 🔗
Geometric Series	$\sum_{n = 0}^{\infty} r^{n}$	$\| r \| < 1$ 🔗	$\| r \| \geq 1$ 🔗	$Sum = \frac{1}{1 - r}$ 🔗
Telescoping Series	$\sum_{n = 1}^{\infty} (b_{n} - b_{n + a})$	$lim_{n \to \infty} b_{n} = L$ 🔗		$Sum = (\sum_{n = 1}^{a} b_{n}) - L$ 🔗
$p$ -Series	$\sum_{n = 1}^{\infty} \frac{1}{(a n + b)^{p}}$	$p > 1$ 🔗	$p \leq 1$ 🔗
Integral Test	$\sum_{n = 0}^{\infty} a_{n}$	$\int_{1}^{\infty} a (n) d n$ converges 🔗	$\int_{1}^{\infty} a (n) d n$ diverges 🔗	$a_{n} = a (n)$ must be continuous 🔗
Direct Comparison	$\sum_{n = 0}^{\infty} a_{n}$	$\sum_{n = 0}^{\infty} b_{n}$ converges and $0 \leq a_{n} \leq b_{n}$ 🔗	$\sum_{n = 0}^{\infty} b_{n}$ diverges and $0 \leq b_{n} \leq a_{n}$ 🔗
Limit Comparison	$\sum_{n = 0}^{\infty} a_{n}$	$\sum_{n = 0}^{\infty} b_{n}$ converges and $lim_{n \to \infty} \frac{a_{n}}{b_{n}} \geq 0$ 🔗	$\sum_{n = 0}^{\infty} b_{n}$ diverges and $lim_{n \to \infty} \frac{a_{n}}{b_{n}} > 0$ 🔗	Also diverges if $lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \infty$ 🔗
Ratio Test	$\sum_{n = 0}^{\infty} a_{n}$	$lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} < 1$ 🔗	$lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} > 1$ 🔗	${a_{n}}$ must be positive 🔗
			Also diverges if 🔗	$lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} = \infty$ 🔗
Root Test	$\sum_{n = 0}^{\infty} a_{n}$	$lim_{n \to \infty} (a_{n})^{1 / n} < 1$ 🔗	$lim_{n \to \infty} (a_{n})^{1 / n} > 1$ 🔗	${a_{n}}$ must be positive 🔗
			Also diverges if 🔗	$lim_{n \to \infty} (a_{n})^{1 / n} = \infty$ 🔗

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