Series Tests

Test For Divergence

If ${\lim }_{n\to \infty }{a}_n$ DNE or if ${\lim }_{n\to \infty }\ne 0$ then the series ${\sum }_{n=1}^{\infty }{a}_n$ is divergent

P-Series

$$\sum _{n=1}^{\infty }\frac{1}{n^p}$$

• $p>1$: converges
• $p\le 1$: diverges
• note: $p=1$ is harmonic series

Integral Test

• Let $a_n=f(n)$
• If $f$ is a continuous, positive, and decreasing function on $[1,\infty )$
• Then then ${\sum }_{n=1}^{\infty }$ has the same convergence as ${\int }_1^{\infty }f(x)dx$

Direct Comparison Test

Suppose that $\sum a_n$ and $\sum b_n$ are series with positive terms

• $\sum b_n$ is convergent and $a_n\le b_n$ for all $n$ $\to \sum a_n$ is convergent
• $\sum b_n$ is divergent and $a_n\ge b_n$ for all $n$ $\to \sum a_n$ is divergent

Limit Comparison Test

Suppose that $\sum a_n$ and $\sum b_n$ are series with positive terms If ${\lim }_{n\to \infty }\frac{a_n}{b_n}=c$

• If $c$ is a finite, positive number, then both series converge or diverge
• $\infty$ is also acceptable when showing divergence
• 0 is also acceptable when showing convergence

Alternating Series Test

If $b_n>0$ and ${\sum }_{n=1}^{\infty }(-1{)}^n{b}_n$ satisfies

1. $b_{n+1}\le b_n$ for all $n\ge N$
2. ${\lim }_{n\to \infty }{b}_n=0$ then the series is convergent

Ratio Test

For the series ${\sum }_{n=1}^{\infty }{a}_n$ if ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|$

• $<1$ absolutely convergent
• $=1$ inconclusive
• $>1$ diverges Good for factorials

Root Test

For the series ${\sum }_{n=1}^{\infty }{a}_n$ if ${\lim }_{n\to \infty }\sqrt[n]{|a_n|}$

• $<1$ absolutely convergent
• $=1$ inconclusive
• $>1$ diverges Good for lots of $n^{th}$ powers