Why does 1/[nlog(n+1)] diverge

[grossgermany]

Homework Statement

Why does the series 1/[nlog(n+1)] diverge

Homework Equations

We know that 1/[nlog(n)] diverges by the integral test. However the question as written does not lend itself to be any integral precisely.

The Attempt at a Solution

 

[Dickfore]

investigate the integral:

$$f(x) = \int_{1}^{x}{\frac{dt}{t \, \log{t}}}$$

as $x \rightarrow \infty$.

 

[grossgermany]

Yes, however the integral doesn't work in this particular case
Note that the question says nlog(n+1), not nlogn

 

[Dick]

Yes, however the integral doesn't work in this particular case
Note that the question says nlog(n+1), not nlogn

1/((n+1)*log(n+1)) diverges, right? You can do an integral test or just notice it's the same series as 1/(n*log(n)) shifted one term. Now try and think of how to use a comparison test.