Question

Determine which of the following series converge or diverge using any theorem or test you like:

1. \[\sum_{k=1}^{\infty} k+2\div k ^{2} +k\]
2. \[\sum_{k=1}^{\infty} k ^{-1}\log(1+k ^{-1})\]
3. \[\sum_{k=1}^{\infty} \log(k)^{2} / \log(2)^{k}\]
4. \[\sum_{k=1}^{\infty} 1/\log ^{k} k\]
5. \[\sum_{k=1}^{\infty} \cot ^{-1} (k)\]
6. \[\sum_{k=5}^{\infty} 1/klog(k)\log(1+\log(1+k))\log(1+\log(1=\log(1+k)))\]
7. 1/2+1/3+1/2^2+1/3^2+1/2^3+1/3^3+1/2^4+1/3^4+....
8. \[\sum_{k=1}^{\infty} (k!)^{2} / (2k)!\]
9. \[\sum_{k=1}^{\infty} (k/(k+1))^{k ^{2}}\]