5.3 The divergence and integral tests  (Page 2/9)

Therefore, for each $k,$ the $k\text{th}$ partial sum $S_k$ satisfies

Since $\underset{k\to \infty }{\text{lim}}\text{ln}(k+1)=\infty ,$ we see that the sequence of partial sums $\left\{S_k\right\}$ is unbounded. Therefore, $\left\{S_k\right\}$ diverges, and, consequently, the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}$ also diverges.

Now consider the series ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^2.$ We show how an integral can be used to prove that this series converges. In [link] , we sketch a sequence of rectangles with areas $1,1\text{/}{2}^2,1\text{/}{3}^2\text{,…}$ along with the function $f(x)=1\text{/}{x}^2.$ From the graph we see that

Therefore, for each $k,$ the $k\text{th}$ partial sum $S_k$ satisfies

We conclude that the sequence of partial sums $\left\{S_k\right\}$ is bounded. We also see that $\left\{S_k\right\}$ is an increasing sequence:

Since $\left\{S_k\right\}$ is increasing and bounded, by the Monotone Convergence Theorem, it converges. Therefore, the series ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^2$ converges.

We can extend this idea to prove convergence or divergence for many different series. Suppose ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ is a series with positive terms $a_n$ such that there exists a continuous, positive, decreasing function $f$ where $f\left(n\right)=a_n$ for all positive integers. Then, as in [link] (a), for any integer $k,$ the $k\text{th}$ partial sum $S_k$ satisfies

Therefore, if ${\displaystyle {\int }_1^{\infty }f\left(x\right)dx}$ converges, then the sequence of partial sums $\left\{S_k\right\}$ is bounded. Since $\left\{S_k\right\}$ is an increasing sequence, if it is also a bounded sequence, then by the Monotone Convergence Theorem, it converges. We conclude that if ${\displaystyle {\int }_1^{\infty }f\left(x\right)dx}$ converges, then the series ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ also converges. On the other hand, from [link] (b), for any integer $k,$ the $k\text{th}$ partial sum $S_k$ satisfies

If $\underset{k\to \infty }{\text{lim}}{\displaystyle {\int }_1^{k+1}f}(x)dx=\infty ,$ then $\left\{S_k\right\}$ is an unbounded sequence and therefore diverges. As a result, the series ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ also diverges. Since $f$ is a positive function, if ${\displaystyle {\int }_1^{\infty }f}(x)dx$ diverges, then $\underset{k\to \infty }{\text{lim}}{\displaystyle {\int }_1^{k+1}f}(x)dx=\infty .$ We conclude that if ${\displaystyle {\int }_1^{\infty }f}(x)dx$ diverges, then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ diverges.