Question

Determining convergence or divergence

Answer 1

\[\sum_{n=1}^{n=\infty} \frac{ 2^n }{ n+1 }\]

Answer 2

Please help

Answer 3

The ratio test is your friend here. If \(a_n=\dfrac{2^n}{n+1}\), then the ratio test tells you that \(\sum a_n\) converges if the following is satisfied:
\[\lim_{n\to\infty}\frac{a_{n+1}}{a_n}<1\]
In this case, you have
\[\lim_{n\to\infty}\frac{2^{n+1}}{n+1+1}\cdot\frac{n+1}{2^n}\]