Example of $\{b_{n}\}$ such that $\sum_{n = 1}^{\infty}b_{n}b_{n + 1} < \infty$ but $\sum_{n = 1}^{\infty}(b_{n + 1} - b_{n})^{2} = \infty$ What is an example of a sequence of positive numbers $\{b_{n}\}$ such that $\sum_{n = 1}^{\infty}b_{n}b_{n + 1} < \infty$ but $\sum_{n = 1}^{\infty}(b_{n + 1} - b_{n})^{2} = \infty$?

Let $b_n=1$ if $n$ is odd, and let $b_n=\dfrac{1}{2^n}$ if $n$ is even.
