For which p series $S=\sum_{n=2}^\infty \log^2{(\sqrt[n]{n})}\frac{\cos{(\frac{n^2 \pi}{n+1})}} {(1+n)^p\log^3{n}}$ converges and absolutely converges?
Convergence: $S=\sum_{n=2}^\infty \frac{1}{n^2}\log^2{n}\frac{(-1)^{n-1}\cos{(\frac{\pi}{n+1})}} {(1+n)^p\log^3{n}}=\sum_{n=2}\frac{(-1)^{n-1}\cos{\frac{\pi}{n+1}}}{n^2(n+1)^p\log{n}}$. This series converges for $p>-2$ by Abel's test? Absolute convergence: $\sum_{n=2}^\infty \frac{\cos{\frac{\pi}{n+1}}}{n^2(n+1)^p\log{n}}$. When does this series converges? Any help is welcome. Thanks in advance.