Complex Functions (Power Series)

[suspenc3]

Hi, Power Series' were not covered in my cal II class, so I don't know how to solve these. Is there a certain way to solve these?

Find the Radiusof convergence and open disk of convergence of the power series:

$$\frac{n^2}{2n+1}(z+6+2i)^n$$

I don't know how to latex the summation but it is there, n=0 - inf.

THanks

 

[quasar987]

See the formula for r here:

http://en.wikipedia.org/wiki/Radius_of_convergence

Note that is you have a function f(z), and you find its power series expansion about a point a, then its radius of convergence is just the distance from a to the nearest singularity of f. You can use this fact to find the radius of convergence of the power series of a real-valued function f(x) by finding the nearest singularity of the corresponding complex valued function f(z).

 

[quasar987]

I should have written "Note that if you have an holomorphic function f(z),..."

I need to refresh my memory about all this...

 

[suspenc3]

So you just take the lim as n->infinity of $$|\frac{C_n}{C_n+1}|$$?

or $$\frac{\frac{n^2}{2n+1}}{\frac{n^2}{2n+1}+1}$$?

 

[quasar987]

That's probably the easiest way.