5.3 The divergence and integral tests  (Page 3/9)

Although convergence of ${\displaystyle {\int }_N^{\infty }f\left(x\right)dx}$ implies convergence of the related series ${\displaystyle \sum _{n=1}^{\infty }{a}_n},$ it does not imply that the value of the integral and the series are the same. They may be different, and often are. For example,

is a geometric series with initial term $a=1\text{/}e$ and ratio $r=1\text{/}e,$ which converges to

However, the related integral ${\displaystyle {\int }_1^{\infty }{(1\text{/}e)}^x}dx$ satisfies

The p -series

The harmonic series ${\displaystyle \sum }_{n=1}^{\infty }1\text{/}n$ and the series ${\displaystyle \sum }_{n=1}^{\infty }1\text{/}{n}^2$ are both examples of a type of series called a p -series.

We know the p -series converges if $p=2$ and diverges if $p=1.$ What about other values of $p\text{?}$ In general, it is difficult, if not impossible, to compute the exact value of most $p$ -series. However, we can use the tests presented thus far to prove whether a $p$ -series converges or diverges.

If $p<0,$ then $1\text{/}{n}^p\to \infty ,$ and if $p=0,$ then $1\text{/}{n}^p\to 1.$ Therefore, by the divergence test,

If $p>0,$ then $f\left(x\right)=1\text{/}{x}^p$ is a positive, continuous, decreasing function. Therefore, for $p>0,$ we use the integral test, comparing

We have already considered the case when $p=1.$ Here we consider the case when $p>0,p\ne 1.$ For this case,

Because

we conclude that

Therefore, ${\displaystyle \sum }_{n=1}^{\infty }1\text{/}{n}^p$ converges if $p>1$ and diverges if $0<p<1.$

In summary,

Estimating the value of a series

Suppose we know that a series ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges and we want to estimate the sum of that series. Certainly we can approximate that sum using any finite sum ${\displaystyle \sum _{n=1}^N{a}_n}$ where $N$ is any positive integer. The question we address here is, for a convergent series ${\displaystyle \sum }_{n=1}^{\infty }{a}_n,$ how good is the approximation ${\displaystyle \sum }_{n=1}^N{a}_n\text{?}$ More specifically, if we let

be the remainder when the sum of an infinite series is approximated by the $N\text{th}$ partial sum, how large is $R_N\text{?}$ For some types of series, we are able to use the ideas from the integral test to estimate $R_N.$