Convergence of series implies convergence of product

Question

Suppose $a_n$ is a positive sequence, then the product $$\prod_{n=1}^{\infty}(1+a_n)$$ converges iff the series $$\sum_{n=1}^{\infty}a_n$$ converges as well.

Any hints please ?

Answer 1

Use the following inequality: $(1+x)\le e^x$

$\prod_{n=1}^{\infty}(1+a_n)\le\prod_{n=1}^{\infty}e^{a_n}=e^{\sum_{n=1}^{\infty}a_n}$

As $\sum_{n=1}^{\infty}a_n$ converges the asked products also converges.