Where is x here?
And the sum should NOT start at 0
It should start at 3 or higher.
Hi there. Well, I was trying to determine the radius and interval of convergence for this power series:
So this is what I did till now:
Then
The thing is this result is clearly wrong, I think the series diverges for any x, there is a discontinuity at n=2 for the sum. But I don't know what I'm doing wrong.
Bye there, thanks for your help!
It does start at zero. x is not needed to get the radius of convergence. It follows from the ratio test for the convergence of series. What I'm trying to find is the radius for the convergence of the power series, and its interval
If exists
Then
The thing is I think it actually doesn't exists the limit I gave before. I mean, the series doesn't converges for any x. But I have to demonstrate it.
There is nothing called RATIO FOR THE CONVERGENCE OF THE POWER SERIES !
Do you mean radius of the convergence or interval of the convergence?
Your series takes the form
First you will compute .
In your problem,
as . (Note that we are dealing with x as a constant while evaluating the limit.).
For the convergence, |x| must be < 1 ,that is -1<x<1
so we reach the form a<x<b , then the radius equal (b-a)/2 = (1-(-1))/2=2/2=1
your radius is 1.
It is impossible for the power series to be divergent for all x.
at least, it must converge for 1 value for x.
and the sum should not start at n=0. (your is not continuous at n=2).