# Thread: Complex power series help

1. ## Complex power series help

Hello everyone,
Firstly, I am new and my apologies if my questions are buried somewhere in the thread heap already.

I am taking a complex analysis course, and well, its 2complex4me! So here are my questions...

I need to find the radius of convergence of the following complex power series: (so naturally z is in C here).

My apologies for not using nice looking math typeset, but I dont know the code for it.

The mth order Bessel function:
Jm(z) = sum[ {((-1)^n) ((z/2)^(m+2n))} / {n!(n+m)!} , n=0, n=infinity ]

Also

sum[ z^n!, n=0, n=inf ]

And

sum[ (n+a^n)z^n, n=0, n=inf ]

I will undoubtedly have more questions as I progress in the course, but if someone can help me out here, I would be very greatful!

Thank you, Damian.

2. Originally Posted by 2complex4me
Hello everyone,
Firstly, I am new and my apologies if my questions are buried somewhere in the thread heap already.

I am taking a complex analysis course, and well, its 2complex4me! So here are my questions...

I need to find the radius of convergence of the following complex power series: (so naturally z is in C here).

My apologies for not using nice looking math typeset, but I dont know the code for it.

The mth order Bessel function:
Jm(z) = sum[ {((-1)^n) ((z/2)^(m+2n))} / {n!(n+m)!} , n=0, n=infinity ]

Also

sum[ z^n!, n=0, n=inf ]

And

sum[ (n+a^n)z^n, n=0, n=inf ]

I will undoubtedly have more questions as I progress in the course, but if someone can help me out here, I would be very greatful!

Thank you, Damian.

Try the ratio test

$\lim_{n_ \to \infty}\left | \frac{a_{n+1}}{a_n}\right |=L$

if L < 1 the series converges absolutely
if L > 1 the series divierges
if L =1 or the limit doesnt exist the test gives no info

He is an example

$\lim_{n \to \infty} \left |\frac{\overbrace{\frac{(-1)^{n+1}(\frac{z}{2})^{m+2(n+1)}}{(n+1)!(n+1+m)!}} ^{a_{n+1}}}{ \underbrace{\frac{(-1)^n (\frac{z}{2})^{m+2n}}{(n)!(n+m)!}}_{a_n}} \right |$

Alot of algebra

$=\lim_{n \to \infty} \left| \frac{z^2}{(1+n)(n+1+m)} \right| =0$

So it converges for all values of z the radius is infinity.

Yeah

Good luck

3. Thanks Empty,
I'll have a crack at that.