# Power series

• May 5th 2010, 11:41 PM
rebecca
Power series
a_nx^n, b_nx^n are two power series with radius of convergence R1, R2, respectively.
If R1 is different than R2, how can i prove that the radius of convergence of the power series (a_n + b_n)x^n is min{R1, R2}?
And what can be said about the radius of cenvergence if R1=R2?

Thanks for any help :)
• May 6th 2010, 04:19 AM
Failure
Quote:

Originally Posted by rebecca
a_nx^n, b_nx^n are two power series with radius of convergence R1, R2, respectively.
If R1 is different than R2, how can i prove that the radius of convergence of the power series (a_n + b_n)x^n is min{R1, R2}?
And what can be said about the radius of cenvergence if R1=R2?

Thanks for any help :)

You cannot prove this because it is not true: The radius of convergence of the series $\sum_n (a_n+b_n)x^n$ might be larger than $\min\{R_1,R_2\}$, but it cannot be strictly smaller.

For a trivial example of this just take $a_n := 1$ and $b_n := -1$ for all $n$. The two series individually have a radius of convergence equal to 1 but the power series with coefficients $a_n+b_n=0$ converges for all $x\in \mathbb{R}$ whatever.

The radius of convergence cannot be strictly smaller, because if $|x|<\min\{R_1,R_2\}$, then the two series $\sum_n a_n x^n,\sum_n b_n x^n$ both converge absolutely and therefore their sum $\sum_n a_n x^n + \sum_n b_n x^n=\sum_n (a_n +b_n)x^n$ also converges absoluely.