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# Convergence Tests

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## Summary of Convergence Tests

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.

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