Determine whether the series converges, and if so, find its sum.

Question

Question

(1) $\sum_{n=1}^{\infty} 3^{-n} 8^{n+1}$

$(2) \sum_{n=2}^{\infty} \frac{1}{n(n-1)}$

(3) $\sum_{n=0}^{\infty}(-3)\left(\frac{2}{3}\right)^{2 n}$

(4) $\sum_{n=1}^{\infty} \frac{1}{e^{2 n}}$

(5) $\sum_{n=1}^{\infty} \ln \frac{n}{n+1}$

(6) $\sum_{n=1}^{\infty}[\arctan (n+1)-\arctan n]$

(7) $\sum_{n=1}^{\infty} \ln \left(\frac{n^{2}+4}{2 n^{2}+1}\right)$

(8) $\sum_{n=1}^{\infty} \frac{1+2^{n}}{3^{n}}$

(9) $\sum_{n=1}^{\infty}\left[\cos \frac{1}{n^{2}}-\cos \frac{1}{(n+1)^{2}}\right]$

Answer 1

Answer #1

Answer 2

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