Determine the convergence of ∑n=1∞1n2+(−1)nn\sum_{n=1}^{\infty} \frac{1}{n^2 + (-1)^n n}∑n=1∞n2+(−1)nn1.
Which approach is valid?
For large nnn, n2−n<n2+(−1)nn<n2+nn^2 - n < n^2 + (-1)^n n < n^2 + nn2−n<n2+(−1)nn<n2+n, so by direct comparison, the series converges.
Since the denominator oscillates, the divergence test shows the series diverges.
By Limit Comparison Test with ∑1n2\sum \frac{1}{n^2}∑n21, the limit is 111, so the series converges.
Since the denominator is not monotone, no comparison test applies.