Convergence Tests

#Calc-2#Math#Series There are multiple tests we can do to test if an infinite series converges.

This page contains the quick guide to each test, each test has its own note going into more detail on how it works and with examples.

An important note, the starting index of a series does not affect convergence

$\sum _{k=1}^{\infty }\frac{1}{k^2}$ converges, so does $\sum _{k=29}^{\infty }\frac{1}{k^2}$ $\sum _{k=1}^{\infty }\frac{1}{k}$ diverges, so does $\sum _{k=1329}^{\infty }\frac{1}{k}$

Geometric Series Convergence §

Useful only for geometric series A geometric series converges if $|r|<1$ in $\sum _{k=1}^{\infty }a{r}^k$

Divergence Test §

This is the $0^{th}$ test, use this one first Divergence Test

If $\sum _{k=1}^{\infty }{a}_k$ converges, then ${\lim }_{k\to \infty }{a}_k=0$ If ${\lim }_{k\to \infty }{a}_k\ne 0$, then the series diverges

This is a one way statement, if $\sum _{k=1}^{\infty }{a}_k$ converges THEN the limit = 0. BUT if the limit = 0, there’s still a chance the series could diverge

A better way of using this test is if the ${\lim }_{k\to \infty }{a}_k\ne 0$, then the series diverges. If it equals 0, the series might converge or diverge.

Integral Test §

Ask “is this easy to integrate” Integral Test

$\sum _{k=1}^{\infty }a(k)$ converges if ${\int }_1^{\infty }a(x)dx$ converges $\sum _{k=1}^{\infty }a(k)$ diverges if ${\int }_1^{\infty }a(x)dx$ diverges

P-Series Test §

$\sum _{k=1}^{\infty }\frac{1}{k^p}$ converges if $p\ge 1$ §

(This is because the associated Integral Test converges if $p>1$)

Direct Comparison Test §

Compare the series to a p-series or a geometric series Comparison Tests

If $a_k\le$ the terms of a convergent series, then the series converges If $a_k\ge$ the terms of a divergent series then the series diverges

Limit Comparison Test §

Compare the series to a p-series or a geometric series Comparison Tests

${\lim }_{k\to \infty }\frac{a_k}{b_k}=c\ge 0$ then both series either converge or both diverge

Alternating Series Test §

Good only with Alternating Series Alternating Series

$\sum _{k=1}^{\infty }(-1{)}^k{a}_k$ converges IF ${\lim }_{k\to \infty }{a}_k=0$ AND $a_k$ is eventually non increasing

Ratio Test §

Very good first test to try, but it can’t handle any kind of p-series Ratio Test

${\lim }_{k\to \infty }|\frac{a_{k+1}}{a_k}|<1$, series converges absolutely (and therefore converges) ${\lim }_{k\to \infty }|\frac{a_{k+1}}{a_k}|>1$, series diverges ${\lim }_{k\to \infty }|\frac{a_{k+1}}{a_k}|=1$, test is inconclusive try another

Root Test §

Really only good to use when there is a k in the exponent Root Test

${\lim }_{k\to \infty }\sqrt[k]{|a_k|}<1$ -> series is convergent ${\lim }_{k\to \infty }\sqrt[k]{|a_k|}>1$ -> series is divergent ${\lim }_{k\to \infty }\sqrt[k]{|a_k|}=1$ -> test is inconclusive