- #1
Pyroadept
- 89
- 0
Homework Statement
Find the radius of convergence of the Taylor series at z = 1 of the function:
[itex]\frac{1}{e^{z}-1}[/itex]
Homework Equations
The Attempt at a Solution
Hi everyone,
Here's what I've done so far.
Multiply top and bottom by minus 1 to get:
-1/(1-e^z)
And then this is the expansion of an infinite geometric series:
1/(1-z) = Ʃ(z^n)
Then use the formula 1/R = lim sup (nth root of absolute value of a_n)
where here a_n is constant, 1.
So then you will get that the lim sup of this is 1.
So then R = 1.
However, I don't like this solution as I think that geometric formula only applies if |z|<1, which is not the case for e^z.
I am a bit stuck as to how else to write out the Taylor series though. And how do I make use of the fact z=1? Does that mean I have to write out a Taylor series in the form Ʃa_n.(z-1)^n?
I'd really appreciate if someone could please point me in the right direction.
Thanks