lean_workbook_plus_47048

We know that for some positive sequence $b_n$ that $\prod_{n=1}^{\infty}(1+b_n)$ converges iff the series $\sum_{n=1}^{\infty} b_n$ converges. This works really well in our favor though because we are already trying to prove the convergence of $\lim_{n\to\infty} a_n=\lim_{n\to\infty} \prod_{i=1}^{n}\left(1+\frac{1}{2^i}\right)=\prod_{i=1}^{\infty}\left(1+\frac{1}{2^i}\right)$ so we can let $b_i=\frac{1}{2^i}$ which means we need to determine the convergence of $\sum_{i=1}^{\infty}\frac{1}{2^i}$ . We see that $\sum_{i=1}^{\infty}\frac{1}{2^i}=\sum_{i=1}^{\infty}\left(\frac{1}{2}\right)^i=1$ by the infinite geometric series formula, so we have that $\prod_{i=1}^{\infty}\left(1+b_i\right)=\prod_{i=1}^{\infty}\left(1+\frac{1}{2^i}\right)$ converges, which is just what we wanted