Mertens’ Theorem

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.

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