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Mathematics
Mathematics
Problem
Evaluate the improper integral
∫
2
∞
1
x
(
ln
x
)
3
d
x
\displaystyle\int_{2}^{\infty} \frac{1}{x(\ln x)^{3}}\,dx
∫
2
∞
x
(
ln
x
)
3
1
d
x
and state what this result implies about the convergence of the series
∑
n
=
2
∞
1
n
(
ln
n
)
3
\displaystyle\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{3}}
n
=
2
∑
∞
n
(
ln
n
)
3
1
.
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