When are these series equal?

Question

Suppose we have a power series $$\sum_{n=0}^\infty {a_nb_nx^n}$$ When is it true that the series obtained by eliminating $b_n$ is proportional to the original series? $$\sum_{n=0}^\infty {a_nx^n}=c\sum_{n=0}^\infty {a_nb_nx^n}\;\;\;\;\;\;\text{for some}\;c\in\mathbb{R}$$

Answer 1

Power series with a positive radius of convergence are equal at points other than the center only if all their coefficients match. So $a_n = c a_n b_n$ for all $n$.

That means for every nonzero value of $a_n$, you have $1 = cb_n$, so $b_n=1/c$. For values of $n$ for which $a_n=0$, the coefficient $b_n$ could be anything.

For example: \begin{align} a & = \text{the sequence } 1,\ 0,\ 2,\ 0,\ 3,\ 0,\ \ldots \\ b & = \text{the sequence } 9, \ 6,\ 9,\ 5,\ 9,\ 7,\ \ldots \\ c & = 1/9 \end{align}