Question

Is the following series convergent?

Answer True/False to the statements.

I found that the limit is 0, and have tried ratio test (L=1) and comparison test, but am unsure how to prove it converges/diverges.

Answer 1

The series:

`sum_(n->oo) 1/(n+sinn)`

is divergent.

Explanation:

As:

`-1 <= sinx <= 1`

we have that:

`1/(n+1) <= 1/(n+sinn) <= 1/(n-1)`

Based on the squeeze theorem we can therefore immediately determine that:

`lim_(n->oo) 1/(n+sinn) = 0`

Consider now the series:

`sum_(n->oo) 1/(n+1)`

As:

`lim_(n->oo) (1/n)/(1/(n+1)) = lim_(n->oo)(n+1)/n = 1`

based in the limit comparison test it must have the same character as the harmonic series that we know to be divergent.

But as:

`1/(n+1) <= 1/(n+sinn)`

then also the series:

`sum_(n->oo) 1/(n+sinn)`

is divergent by direct comparison.