Question

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

Answer 1

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.