product of the terms of two series that diverges to $\infty$

Question

Suppose that $0\leq p_n\leq 1$ for each $n$. Also suppose $\sum_{n=1}^\infty p_n = \infty$ and $\sum_{n=1}^\infty (1-p_{n}) = \infty$. How can you prove $\sum_{n=1}^\infty p_n (1 - p_{n+1}) = \infty$?

Answer 1

This answer assumes that $0\le p_n\le 1$ for all $n$. If you want to solve this problem without that assumption, this solution won't work.

Split into three cases:

1. $p_n\ge\frac12$ from some point $n_0$ onward. Then $\sum_{n=n_0}^\infty p_n(1-p_{n+1}) \ge \frac12 \sum_{n=n_0}^\infty (1-p_{n+1}) = \infty$.
2. $p_n\le\frac12$ from some point $n_0$ onward. Then $\sum_{n=n_0}^\infty p_n(1-p_{n+1}) \ge \frac12 \sum_{n=n_0}^\infty p_n = \infty$.
3. Neither of the above hold. Then there are infinitely many $n_j$ such that $p_{n_j} \ge \frac12$ and $p_{n_j+1} \le \frac12$. Then $\sum_{n=1}^\infty p_n(1-p_{n+1}) \ge \sum_{j=1}^\infty p_{n_j}(1-p_{n_j+1}) \ge \sum_{j=1}^\infty \frac14 = \infty$.