How do you use the Integral test on the infinite series #sum_(n=1)^oo1/root5(n)# ?

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Answer 1 · 2014-09-17T09:47:41

Since the integral
#int_1^oo 1/root5x dx#
diverges, we can conclude that the series
#sum_{n=1}^oo 1/root5n#
also diverges by Integral Test.

Here is the details of the evaluation of the integral.

#int_1^infty 1/root5x dx#

#=lim_{t to infty}[5/4x^{4/5}]_1^{t}#

#= {5}/{4} lim_{t to infty}(t^{4/5}-1)#

#=5/4(infty-1)=infty#