Find Taylor's polynomial of the following function in : .
Determine the radius of convergence of the series.
My attempt :
. Which answers the first question.
Now trying to figure out the second question :
So .
I apply the ratio test : which tends to when tends to .
Hence the radius of convergence is ...? It makes no sense at all.
I'd like to know where is(are) my error(s). Thanks a lot!
I made a mistake, I've edited my post. However I still don't reach the Taylor series you provided. I don't see why the series I found is not a Taylor series.
I've used the fact that .
I've found out that .
Now you make me doubt the formula holds for the function due to the singularity at .
Edit : I understand the equality you gave, but I don't know how to reach it. I mean I don't know how to reach that the Taylor Series is
Edit 2 : Sorry, I just found my error. I forgot to evaluate in ...
I think I'll do it. Thanks a lot!
The radius of convergence of a function will extend to the nearest point at which the function is not analytic. Here, the function is which is not analytic at z= 0. Since the center of the region of convergence is at , the radius of convergence is |0-(-1)|= 1.
This can be used even for real functions. For example, the function is analytic for all x except -i or -i. If it is expanded in a power series about x= 0, then its radius of convergence is |0- i|= 1.
If it is expanded in a power series about x= 1, its radius of convergence is .