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Summary of Convergence Tests

Last updated

Dec 21, 2020
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- 3-D Coordinate Systems
- Summary of Theorems

This page is a draft and is under active development.

( \newcommand{\kernel}{\mathrm{null}\,}\)

Summary of Convergence Tests
Series or Test	Conclusions	Comments
Divergence Test For any series $\sum_{n = 1}^{\infty} a_{n}$ , evaluate $lim_{n \to \infty} a_{n}$ .	If $lim_{n \to \infty} a_{n} = 0$ , the test is inconclusive.	This test cannot prove convergence of a series.
	If $lim_{n \to \infty} a_{n} \neq 0$ , the series diverges.	This test cannot prove convergence of a series.
Geometric Series $\sum_{n = 1}^{\infty} a r^{n - 1}$	If $\| r \| < 1$ , the series converges to $a / (1 - r)$ .	Any geometric series can be reindexed to be written in the form $a + a r + a r^{2} + \dots$ , where $a$ is the initial term and r is the ratio.
Geometric Series $\sum_{n = 1}^{\infty} a r^{n - 1}$	If $\| r \| \geq 1,$ the series diverges.
p-Series $\sum_{n = 1}^{\infty} \frac{1}{n^{p}}$	If $p > 1$ , the series converges.	For $p = 1$ , we have the harmonic series $\sum_{n = 1}^{\infty} 1 / n$ .
p-Series $\sum_{n = 1}^{\infty} \frac{1}{n^{p}}$	If $p \leq 1$ , the series diverges.
Comparison Test For $\sum_{n = 1}^{\infty} a_{n}$ with nonnegative terms, compare with a known series $\sum_{n = 1}^{\infty} b_{n}$ .	If $a_{n} \leq b_{n}$ for all $n \geq N$ and $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.	Typically used for a series similar to a geometric or $p$ -series. It can sometimes be difficult to find an appropriate series.
	If $a_{n} \geq b_{n}$ for all $n \geq N$ and $\sum_{n = 1}^{\infty} b_{n}$ diverges, then $\sum_{n = 1}^{\infty} a_{n}$ diverges.
Limit Comparison Test For $\sum_{n = 1}^{\infty} a_{n}$ with positive terms, compare with a series $\sum_{n = 1}^{\infty} b_{n}$ by evaluating $L = lim_{n \to \infty} \frac{a_{n}}{b_{n}} .$	If $L$ is a real number and $L \neq 0$ , then $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ both converge or both diverge.	Typically used for a series similar to a geometric or $p$ -series. Often easier to apply than the comparison test.
	If $L = 0$ and $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.
	If $L = \infty$ and $\sum_{n = 1}^{\infty} b_{n}$ diverges, then $\sum_{n = 1}^{\infty} a_{n}$ diverges.
Integral Test If there exists a positive, continuous, decreasing function $f$ such that $a_{n} = f (n)$ for all $n \geq N$ , evaluate $\int_{N}^{\infty} f (x) d x .$	$\int_{N}^{\infty} f (x) d x$ and $\sum_{n = 1}^{\infty} a_{n}$ both converge or both diverge.	Limited to those series for which the corresponding function f can be easily integrated.
Alternating Series $\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} b_{n}$ or $\sum_{n = 1}^{\infty} {(- 1)}^{n} b_{n}$	If $b_{n + 1} \leq b_{n}$ for all $n \geq 1$ and $b_{n} \to 0$ , then the series converges.	Only applies to alternating series.
Ratio Test For any series $\sum_{n = 1}^{\infty} a_{n}$ with nonzero terms, let $ρ = lim_{n \to \infty} ∣ \frac{a_{n + 1}}{a_{n}} ∣$	If $0 \leq ρ < 1$ , the series converges absolutely.	Often used for series involving factorials or exponentials.
	If $ρ > 1$ or $ρ = \infty$ , the series diverges.
	If $ρ = 1$ , the test is inconclusive.
Root Test For any series $\sum_{n = 1}^{\infty} a_{n}$ , let $ρ = lim_{n \to \infty} \sqrt[n]{\| a_{n} \|}$ .	If $0 \leq ρ < 1$ , the series converges absolutely.	Often used for series where $\| a_{n} \| = b_{n}^{n}$ .
	If $ρ > 1$ or $ρ = \infty$ , the series diverges.
	If $ρ = 1$ , the test is inconclusive.