# To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

I am facing difficulty in this problem. Help Needed!

• Do you know a formula to express $R$ in terms of $(a_n)$? – Michael M Jul 28 '15 at 1:14
• Would something like: $b_n = \begin{cases}1 & n \le N \\ 2^{-n} & n > N \end{cases}$ be allowed, or do you consider this "essentially the same" as the series $b_n = 2^{-n}$? – JimmyK4542 Jul 28 '15 at 1:14
• Any sequence that has $\lim\limits_{n\to\infty}\frac{b_{n+1}}{b_n} = \frac{1}{2}$ will do. – Winther Jul 28 '15 at 1:24