1. ## Addition of power series

My exam of Analysis is getting closer and closer and therefore this is rather urgent.

In class we talked about power series and the radius of convergence. After that we talked about the addition of two power series with radii of convergence respectively Ra and Rb. Now the theorem says that Σ(an+bn)x^n has a radius of convergence R>=min{Ra,Rb}. Finally, there are examples of such power series where R is strictly greater than min{Ra,Rb}. My question is: which example of such power series?

Finding such example is quite important, it is asked in my book with exclamation marks..

Leslon

2. Originally Posted by Leslon
My exam of Analysis is getting closer and closer and therefore this is rather urgent.
You mean to say your analysis exam is converging to the date.

In class we talked about power series and the radius of convergence. After that we talked about the addition of two power series with radii of convergence respectively Ra and Rb. Now the theorem says that Σ(an+bn)x^n has a radius of convergence R>=min{Ra,Rb}. Finally, there are examples of such power series where R is strictly greater than min{Ra,Rb}. My question is: which example of such power series?

Finding such example is quite important, it is asked in my book with exclamation marks..

Leslon
Take for example, $\displaystyle a_n = 1$ and $\displaystyle b_n = -1$. Both have radius of convergence equal to $\displaystyle 1$ yet their sum has infinite radius of convergence.

3. First of all, thank you for replying so fast..

I understand that both radii of $\displaystyle a_n$ and $\displaystyle b_n$ are 1. But why is the radius of convergence infinite? If I try to find the radius according to the following theorem, i get $\displaystyle /frac{0}{0}$

lim $\displaystyle \frac{|c_n|}{|c_{n+1}|} = R$ if the limit exists or in special case +∞

So the question is how you got infinite as the radius of convergence of the sum of the power series?

Leslon

PS: And yes, my exam is converging rapidly to that date

EDIT: I understand now and thank you very much for your help!!

4. Hm... Hello ^^

The power series would be $\displaystyle \sum_{n \ge 0} 0 \cdot x^n=0$

So the sum is finite and converges to 0, $\displaystyle \forall x \in \mathbb{R}$, which makes the radius of convergence (remember, the invertal in which the series converges) infinite ~