Chapter 9.5, Problem 52E

To determine

To find the function is converges or diverges.

The series converges.

Answer to Problem 52E

The series converges.

Explanation of Solution

Given:

The given $N=1$

Calculation:

For $N=1$ , $n\ge 1$ , then the sum $a_1+a_2+a_3+\mathrm{..........}+a_n$ provides an upper bound for ${\displaystyle {\int }_1^{n+1}f(x)dx}$ and sum $a_1+a_2+a_3+\mathrm{..........}+a_n$ provides a lower bound for ${\displaystyle {\int }_1^{n}f(x)dx}$ ,

  ${\displaystyle {\int }_1^{n+1}f(x)dx}<a_1+a_2+a_3+\mathrm{..........}+a_n<{\displaystyle {\int }_1^{n}f(x)dx}$ .

If the integral diverges, it must go to infinity and the first inequality forces the partial sums of the series to go to infinity as well, so the series is divergent. If the integral converges, then the second inequality puts an upper bound on the partial sums of the series, and since they are a non-decreasing sequence, they must converge to a finite sum for the series.