Chapter 10.5, Problem 7E

To determine

To find: that the whether the series converges or diverges.

Answer to Problem 7E

Diverges

Explanation of Solution

Given:

  ${\displaystyle \sum _{n=1}^{\infty }\frac{5}{n+1}}$

Calculation:

  ${\displaystyle \sum _{n=1}^{\infty }5/\left(n+1\right)}$

The given series

  $f\left(x\right)=5/\left(x+1\right)$

Rewrite the series.

  ${\displaystyle {\int }_1^{\infty }5/\left(x+1\right)dx}=\underset{k\to \infty }{\lim }{\displaystyle {\int }_1^{\infty }5/\left(x+1\right)}$

Replace $\infty$ with the limit

  $\underset{k\to \infty }{\lim }{\left[5\ln \left(x+1\right)\right]}_1^{\infty }$

The integral of 1/x is ln(x). Similarly, the integral of 5/(1+x) is 5ln (1+x).

  $\underset{k\to \infty }{\lim }\left[5\ln \left(k+1\right)-5\ln \left(1+1\right)\right]$

Substitute in the limits.

  $\underset{k\to \infty }{\lim }\left(5\ln \left(\infty +1\right)+5\ln \left(2\right)\right)=\infty$

  $\underset{k\to \infty }{\lim }5\ln \left(k+1\right)$ is infinitely large.