Question

Use the integral test to determine whether the infinite series is convergent. ∑n=1∞n−13 fill in the corresponding integrand and the value of the improper integral. enter inf for ∞, -inf for −∞, and dne if the limit does not exist. compare with ∫∞1 dx = by the integral test, the infinite series ∑n=1∞n−13
a. converges
b. diverges

Answer 1

Looks like the series is

$\displaystyle\sum_{n=1}^\infty n^{-13}$

We have

$\displaystyle\sum_{n=1}^\infty\frac1{n^{13}}$
$\displaystyle=-\frac1{12x^{12}}\bigg|_{x=1}^{x\to\infty}$
$\displaystyle=-\frac1{12}\left(\lim_{x\to\infty}\frac1{x^{12}}-1\right)=\frac1{12}$

Therefore the sum converges, since its value is smaller than $\dfrac1{12}$.