Suppose $0 \le a_n \le 2^n \le b_n$ and $0 \le c_n \le 2^{-n} \le d_n$ for all $n \ge 1$. Of the series, \begin{align*} \sum_{n=1}^{\infty} a_n, \quad \sum_{n=1}^{\infty} b_n, \quad \sum_{n=1}^{\infty} c_n, \quad \sum_{n=1}^{\infty} d_n,\end{align*} which will definitely converge and which will definitely diverge? Explain.
