series help!

[clw89]

determine whether the series is convergent or divergent:

the series from n=1 to infinity of ln(n)/n[sup:36no4ihc]3[/sup:36no4ihc]

Thanks!!!

 

[galactus]

Try using the integral test.

$\displaystyle \int_{1}^{L}\frac{ln(n)}{n^{3}}=\frac{-ln(L)}{2L^{2}}-\frac{1}{4L^{2}}+\frac{1}{4}$

Now, take the limit:

$\displaystyle \lim_{L\to\infty}\left[\frac{-ln(L)}{2L^{2}}-\frac{1}{4L^{2}}+\frac{1}{4}\right]$

See the limit?. What does that tell you about convergence?.

 

[pka]

Because $\displaystyle \ln(n) < n$ direct comparison is faster.