Power Series – How to Determine the Radius of Convergence of a Power Series

Question

Given two power series $$\sum_{n=0}^{\infty} a_nx^n, \sum_{n=0}^{\infty} b_nx^n$$ with convergent radius $R_{1}$ and $R_{2}$ respectively. Suppose $R_{1}<R_{2}$,now what about the convergent radius of $\sum_{n=0}^{\infty} (a_n+b_n)x^n$

Answer 1

For a power series $\sum_n c_nx^n$, the radius of convergence is $\sup\{R>0, \{c_nx^n\}\mbox{ is bounded if }|x|\leq R\}$.

Let $x$ such that $|x|<R_1$. Then $|(a_n+b_n)x^n|\leq |a_n||x|^n+|b_n||x^n|$ and the sequence $\{(a_n+b_n)x^n\}$ is bounded. Hence the radius of convergence of $\sum_n(a_n+b_n)x^n$ is $\geq R_1$. If we take $|x|\in (R_1,R_2)$, then $\{(a_n+b_n)x^n\}$ is not bounded, as a sum of a bounded sequence with an unbounded one. So the radius of convergence is at most $R_1$.

Conclusion: the radius of convergence of $\sum_n (a_n+b_n)x^n$ is $R_1$.

Now we look at the case $R_1=R_2$. Let $R>R_1$ and $\sum_nc_nx^n$ a series or radius of convergence $R$. By the previous case, $\sum_n(-a_n+c_n)x^n$ has a radius of convergence $R_1$ but $\sum_n(a_n+(-a_n+c_n))x^n$ has a radius of convergence $R$. So the sum of two series of radius of convergence $R_1$ can be of radius of convergence $R\geq R_1$.