sum, n=1,infty, of 1/(n+nsin^2n) converges? (plz check ans)

[njmiano]

I just took a test and I am not to sure about this question

converge or diverge: (sum of a series n = 1 to infinite) 1/ (n+n sin^2 n)

I factored out an n, and made it 1 / (n (1 + sin^2 n)... then I showed that this sum (when large enough) is less than 1 / n^2 which converges.

Is this right, or is logic against me on this one?

 

[Deleted member 4993]

Re: Did I get this question right? (infinite series)

I just took a test and I am not to sure about this question
(sum of a series n = 1 to infinite) 1/ (n+n sin^2 n)
converge or diverge?
I factored out an n, and made it 1 / (n (1 + sin^2 n)... then I showed that this sum (when large enough) is less than 1 / n^2

For that to happen you would have (1+sin^2 n) > n

That is not correct

which converges.
Is this right, or is logic against me on this one?

 

[fasteddie65]

Re: Did I get this question right? (infinite series)

0 ? sin^2 n ? 1
1 ? 1 + sin^2 n ? 2

1/2 ? 1/(1 + sin^2 x n) ? 1

1/(2n) ? 1/[n(1 + sin^2 x n)] ? 1/n

Since {1/(2n)} diverges -- it is a multiple of {1/n} which diverges, so does this series, by the Comparison Test.