# Answer on Real Analysis Question for Divya wadhwa

Question #17566

Determine whether the series converges.

summation[1/{n+(-1)^n}^2]

summation[1/{n+(-1)^n}^2]

Expert's answer

Let us show that the series converges.

It is well known that the series summation[ 4/n^2] converges

Since n/2 <n+(-1)^n for n>2, we get that 1/{n+(-1)^n}^2 < 4/n^2, and so

summation[1/{n+(-1)^n}^2] < summation[ 4/n^2 ] < infinity

Hence the series summation[1/{n+(-1)^n}^2] converges.

It is well known that the series summation[ 4/n^2] converges

Since n/2 <n+(-1)^n for n>2, we get that 1/{n+(-1)^n}^2 < 4/n^2, and so

summation[1/{n+(-1)^n}^2] < summation[ 4/n^2 ] < infinity

Hence the series summation[1/{n+(-1)^n}^2] converges.

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