How do you use the Integral test on the infinite series $sum_(n=1)^oo1/n^5$ ?

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Answer 1 · 2014-09-24T23:46:19

By Integral Test,

$sum_{n=1}^infty 1/n^5$ converges.

Let us look at some details.

Let us evaluate the corresponding improper integral.

$int_1^infty 1/x^5 dx$

$=lim_{t to infty}int_1^tx^{-5} dx$

$=lim_{t to infty}[x^{-4}/-4]_1^t$

$=-1/4lim_{t to infty}[1/x^4]_1^t$

$=-1/4 lim_{t to infty}[1/t^4-1]$

$=-1/4(0-1)=1/4$

Since the integral

$int_1^infty 1/x^5 dx$

converges to $1/4$ ,

$sum_{n=1}^infty 1/n^5$

also converges by Integral Test.

How do you use the Integral test on the infinite series sum_(n=1)^oo1/n^5sum_(n=1)^oo1/n^5 ?