## Mertens’ Theorem

Let $(a_n)$ and $(b_n)$ be real or complex sequences.

If the series $\sum_{n=0}^\infty a_n$ converges to $A$ and $\sum_{n=0}^\infty b_n$ converges to $B$, and at least one of them converges absolutely, then their Cauchy product converges to $AB$.

An immediate corollary of Mertens’ Theorem is that if a power series $f(x)=\sum a_kx^k$ has radius of convergence $R_a$, and another power series $g(x)=\sum b_kx^k$ has radius of convergence $R_b$, then their Cauchy product converges to $f\cdot g$ and has radius of convergence at least the minimum of $R_a, R_b$.

Note that a power series converges absolutely within its radius of convergence so Mertens’ Theorem applies.