Summary of Convergence Tests

Summary of Convergence Tests

Series or Test	Conclusions	Comments
Divergence Test For any series $\sum^∞_{n=1}a_n$, evaluate $\lim_{n→∞}a_n$.	If $\lim_{n→∞}a_n=0$, the test is inconclusive.	This test cannot prove convergence of a series.
	If $\lim_{n→∞}a_n≠0$, the series diverges.
Geometric Series $\sum^∞_{n=1}ar^{n−1}$	If $|r|<1$, the series converges to $a/(1−r)$.	Any geometric series can be reindexed to be written in the form $a+ar+ar^2+⋯$, where $a$ is the initial term and r is the ratio.
	If $|r|≥1,$ the series diverges.
p-Series $\sum^∞_{n=1}\frac{1}{n^p}$	If $p>1$, the series converges.	For $p=1$, we have the harmonic series $\sum^∞_{n=1}1/n$.
	If $p≤1$, the series diverges.
Comparison Test For $\sum^∞_{n=1}a_n$ with nonnegative terms, compare with a known series $\sum^∞_{n=1}b_n$.	If $a_n≤b_n$ for all $n≥N$ and $\sum^∞_{n=1}b_n$ converges, then $\sum^∞_{n=1}a_n$ converges.	Typically used for a series similar to a geometric or $p$-series. It can sometimes be difficult to find an appropriate series.
	If $a_n≥b_n$ for all $n≥N$ and $\sum^∞_{n=1}b_n$ diverges, then $\sum^∞_{n=1}a_n$ diverges.
Limit Comparison Test For $\sum^∞_{n=1}a_n$ with positive terms, compare with a series $\sum^∞_{n=1}b_n$ by evaluating $L=\lim_{n→∞}\frac{a_n}{b_n}.$	If $L$ is a real number and $L≠0$, then $\sum^∞_{n=1}a_n$ and $\sum^∞_{n=1}b_n$ 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 $\sum^∞_{n=1}b_n$ converges, then $\sum^∞_{n=1}a_n$ converges.
	If $L=∞$ and $\sum^∞_{n=1}b_n$ diverges, then $\sum^∞_{n=1}a_n$ diverges.
Integral Test If there exists a positive, continuous, decreasing function $f$ such that $a_n=f(n)$ for all $n≥N$, evaluate $∫^∞_Nf(x)dx.$	$∫^∞_Nf(x)dx$ and $\sum^∞_{n=1}a_n$ both converge or both diverge.	Limited to those series for which the corresponding function f can be easily integrated.
Alternating Series $\sum^∞_{n=1}(−1)^{n+1}b_n$ or $\sum^∞_{n=1}(−1)^nb_n$	If $b_{n+1}≤b_n$ for all $n≥1$ and $b_n→0$, then the series converges.	Only applies to alternating series.
Ratio Test For any series $\sum^∞_{n=1}a_n$ with nonzero terms, let $ρ=\lim_{n→∞}∣\frac{a_{n+1}}{a_n}∣$	If $0≤ρ<1$, the series converges absolutely.	Often used for series involving factorials or exponentials.
	If $ρ>1$ or $ρ=∞$, the series diverges.
	If $ρ=1$, the test is inconclusive.
Root Test For any series $\sum^∞_{n=1}a_n$, let $ρ=\lim_{n→∞}\sqrt[n]{|a_n|}$.	If $0≤ρ<1$, the series converges absolutely.	Often used for series where $|a_n|=b^n_n$.
	If $ρ>1$ or $ρ=∞$, the series diverges.
	If $ρ=1$, the test is inconclusive.