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.