Question

Use the ratio test to determine whether each series is convergent or divergent.
a) 2/5 + 4/10 + 8/15 + ...

Answer 1

\(a_n=\frac{2^n}{5n}\), so the ratio test is:
\[\lim_{n\rightarrow \infty}\left|\frac{\frac{2^{n+1}}{5(n+1)}}{\frac{2^n}{5n}}\right|\\
\lim_{n\rightarrow \infty}\left|\frac{2^{n+1}\cdot 5n}{2^n\cdot 5(n+1)}\right|\\
\lim_{n\rightarrow\infty}\frac{2n}{n+1}\\
2 > 1\]
Hence the series diverges.