Compute the value of the following improper integral

Question

Compute the value of the following improper integral

∫(5(ln(x)))/x^(8)dx, from 1 to infinity =

Does the series ∑ (5(ln(n)))/(n^(8))dx, from n=1 to infinity converge or diverge?

Hint:


Use integration by parts.

Answer 1

u = ln x

 

du = dx/x

 

dv = x^-8dx

 

v = -x^-7/7

 

-(1/7)x^-7*ln x - ∫-(1/7)x^-8 =

 

-(1/7)x^-7*ln x - (1/49)x^-7  (1, ∞)

 

0 - (-1/49) = 1/49

 

The series will converge. This can be treated like a p-series, and since p in this case is 8 (which is greater than 1) the p-series will converge.