5.3 The divergence and integral tests  (Page 4/9)

We illustrate [link] in [link] . In particular, by representing the remainder $R_N=a_{N+1}+a_{N+2}+a_{N+3}+\text{⋯}$ as the sum of areas of rectangles, we see that the area of those rectangles is bounded above by ${\displaystyle {\int }_N^{\infty }f\left(x\right)dx}$ and bounded below by ${\displaystyle {\int }_{N+1}^{\infty }f\left(x\right)dx}.$ In other words,

and

We conclude that

Since

where $S_N$ is the $N\text{th}$ partial sum, we conclude that

Key concepts

• If $\underset{n\to \infty }{\text{lim}}{a}_n\ne 0,$ then the series ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ diverges.
• If $\underset{n\to \infty }{\text{lim}}{a}_n=0,$ the series ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ may converge or diverge.
• If ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ is a series with positive terms $a_n$ and $f$ is a continuous, decreasing function such that $f\left(n\right)=a_n$ for all positive integers $n,$ then
  
  ${\displaystyle \sum }_{n=1}^{\infty }{a}_n\text{and}{\displaystyle {\int }_1^{\infty }f\left(x\right)dx}$
  
  either both converge or both diverge. Furthermore, if ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ converges, then the $N\text{th}$ partial sum approximation $S_N$ is accurate up to an error $R_N$ where ${\displaystyle {\int }_{N+1}^{\infty }f\left(x\right)dx<R_N<{\displaystyle {\int }_N^{\infty }f\left(x\right)dx}}.$
• The p -series ${\displaystyle \sum }_{n=1}^{\infty }1\text{/}{n}^p$ converges if $p>1$ and diverges if $p\le 1.$

Key equations

• Divergence test
  $\text{If}{a}_n\nrightarrow 0\text{as}n\to \infty ,{\displaystyle \sum _{n=1}^{\infty }{a}_n}\text{diverges}.$
• p -series
  ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^p}}\{\begin{array}{l}\text{converges if}p>1\\ \text{diverges if}p\le 1\end{array}$
• Remainder estimate from the integral test
  ${\displaystyle {\int }_{N+1}^{\infty }f(x)dx}<R_N<{\displaystyle {\int }_N^{\infty }f(x)dx}$

For each of the following sequences, if the divergence test applies, either state that $\underset{n\to \infty }{\text{lim}}{a}_n$ does not exist or find $\underset{n\to \infty }{\text{lim}}{a}_n.$ If the divergence test does not apply, state why.

State whether the given $p$ -series converges.