I'm not sure why that wouldn't converge at z=1. As k increases infinitely, shouldn't 1/k become infinitely small?
Isn't saying that like saying that the sequence:
(\frac{1}{n})^{∞}_{n=1}
doesn't converge? Or does it actually not?
Well when z=-2, we get:
∞
\sum1k2=1+1+...+1=∞ as k→∞
k=0
So ak does not tend toward zero.
Then at z=-3+i:
∞
\sumik2
k=0
So again, ak does not tend toward zero.
In general, we can see that for any z such that z+3≥1, ak will not approach 0 as k increases towards infinity. That's a huge help...
Well that's partially why I used this example. I'm not sure how to handle when the exponent on the complex number is not just k. I realize now I made a mistake in the original post (I didn't make it on my problem set). If I'm right and the radius is 1, then what I meant to write was that z=-2...
Homework Statement
I have a problem set that asks me to determine, first, the radius of convergence of a complex series (using the limit of the coefficients), and second, whether or not the series converges anywhere on the radius of convergence.
Homework Equations
As an example:
Σ(z+3)k2
with...
Homework Statement
Prove that \varphi(n)→∞ as n→∞. n\inZ
Homework Equations
\varphi(n) = the number of integers less than n that are coprime to n.
The Attempt at a Solution
My professor said that we need to show that \varphi(n) is always greater than some increasing estimate of...