Question: Determine whether the series is convergent or divergent. If it is convergent, find its sum.

1. \(\sum_{n=1}^{\infty} \frac{10^{n}}{(-9)^{n-1}}\)
2. \(\sum_{n=1}^{\infty} \frac{n+1}{2 n-3}\)
3. \(\sum_{n=1}^{\infty} \frac{2}{n^{2}+4 n+3}\)
4. \(\sum_{n=1}^{\infty} \frac{e^{n}}{n^{2}}\)
5. \(\sum_{n=1}^{\infty} \frac{1+2^{n}}{3^{n}}\)
6. \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)}\)

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