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.

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1 Answer
Jun 13, 2018

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.