Here is my problem:
Let sum(a_k*z^k, k=0->infinity)
be the power series expansion of
2/((3-2z)^2).
I need to figure out a closed form expression for a_k.
My first problem is figuring out how to manipulate the function into something I know how to express (geometric series?).
Secondly (although this will probably be easier once I figure out issue number 1) is that I don't see how it will be obvious to express a_k unless you just explicitly get the power series.
any hints?
Very nice approach. Uniform continuity is great!
Here is one more I am stuck on, if you have the time:
f(z)=sinz/(pi-z)
g(z)=(z(pi-z))/sinz
I need to find the power series representation of f(z) at pi, and its radius of convergence. Then I need to find the radius of convergence of the power series of g(z) at pi/2.
I can write out sinz as a power series no problem. And again I feel like I should write out the denominator as a geometric series, and then multiply the two series together. I can't figure out how to express it as a geometric series.
If are positive real numbers then we have the exponents rules:
If are (non-zero) complex numbers then we may ask if:
It turns out that #1 is always true. But #2 needs to be used with a lot of care. One way when #2 always works is when . You certainly seen that . Well, that is another special case when is an integer.