Question: Use an appropriate testing method to determine if the series converges or diverges. If the series is alternating and convergent, then determine if it converges absolutely or conditionally. Please state tests you are using!

1. \(\sum_{n=1}^{\infty} \frac{(2 n+1)^{n}}{n^{2 n}}\)
2. \(\sum_{n=1}^{\infty} \frac{\sqrt{n^{2}-1}}{n^{3}+2 n^{2}+5}\)
3. \(\sum_{k=1}^{\infty} \frac{2^{k} k !}{(k+2) !}\)
4. \(\sum_{n=1}^{\infty} \frac{\sin 2 n}{1+2^{n}}\)
5. \(\sum_{n=2}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}-1}[3]\)
6. \(\sum_{n=1}^{\infty} n \sin \left(\frac{1}{n}\right)[1]\)
7. \(\sum_{k=1}^{\infty} \frac{5^{k}}{3^{k}+4^{k}}\)
8. \(\sum_{n=1}^{\infty} n^{2} e^{-n^{3}}\)

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