Comparison Tests for Convergence Practice Test

Since $\frac{2^n}{3^n+n}\le \frac{1}{n}$ for $n\ge1$ and $\sum \frac{1}{n}$ diverges, the series diverges.

Since $\frac{2^n}{3^n+n}\le \left(\frac{2}{3}\right)^n$ for $n\ge1$ and $\sum \left(\frac{2}{3}\right)^n$ converges, the series converges.

Since $\frac{2^n}{3^n+n}\le \left(\frac{3}{2}\right)^n$ for $n\ge1$ and $\sum \left(\frac{3}{2}\right)^n$ diverges, the series converges.

Since $\frac{2^n}{3^n+n}\ge \frac{1}{n}$ for $n\ge1$ and $\sum \frac{1}{n}$ diverges, the series converges.

Since $\frac{2^n}{3^n+n}\ge \left(\frac{2}{3}\right)^n$ for $n\ge1$ and $\sum \left(\frac{2}{3}\right)^n$ converges, the series diverges.