Hello, quick question:
Suppose I am to multiply two absolute convergent series and already know the convergence radius of both: it is r=1 with the convergence interval being [-1,1]. Now, what happens for this radius if I look at whatever comes out of the multiplication? Does the convergence radius change somehow because of the multiplication or does it stay r=1?
>>8115056
the radius of convergence might be bigger than 1.
You know it's at least 1 easily, but if you take the series for sqrt(1-x) and multiply it by itself, you get 1-x which has an infinite radius of convergence.
>>8115070
Oh, that makes things difficult. But thanks, now I won't make a mistake.
>>8115056
Okay OP here again, I think I would need more help.
I need to find the series representation of [1/(1-x)]*[1/(1+x^2)] and state the convergence interval.
Now I have found the closed forms of both terms, they are sum k=0 to inf x^k and sum k=0 to inf (-1)^k*x^(2k). Both have the convergence radius r=1.
So what I do now? I thought about taking the cauchy product, but this is quite difficult and leads me to nowhere. Any tips on what to do?
>>8115107
your reflex should be a partial fraction decomposition
[1/(1-x)]*[1/(1+x^2)] = 0.5 * [1/(1-x)] + 0.5 * [(1+x)/(1+x^2)]
you know the series of 1/(1-x), and you can easily find the series of (1+x)/(1+x^2) from the series of 1/(1+x^2)
you then just sum everything to obtain the general term.
You can then start looking for the radius of convergence
>>8115122
Interesting approach, I haven't seen this used on series yet. I will try it this way, thanks!
>>8115141
np
I think you're at a point where you have to become "creative" and use everything you have learned in the past instead of only the current lesson, aren't you?
>>8115149
Yes, this is quite true. I would never have thought about partial fractions on my own. This is an important lesson.
>>8115070
that's not how you multiply series
>>8115444
this is going to be good. What was wrong about that?
