Complex Analysis: Radius of Convergence

In summary, the problem is asking to find the radius of convergence for two power series: a) \sum z^{n!} and b) \sum (n+2^{n})z^{n}. To find the radius of convergence, we can use the formula Radius = 1/(limsup n=>infinity |cn|^1/n). For part a), cn is just 1, so the radius of convergence is 1. For part b), cn = n+2^{n}, and we can use the limsup to find the radius of convergence, which turns out to be 1/2. It is important to be familiar with different methods, such as the ratio test, for finding the radius of convergence.
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Homework Statement


Find the radius of convergence of the power series:
a) [tex]\sum[/tex] z[tex]^{n!}[/tex]
n=0 to infinity

b) [tex]\sum[/tex] (n+2[tex]^{n}[/tex])z[tex]^{n}[/tex]
n=0 to infinity

Homework Equations


Radius = 1/(limsup n=>infinity |cn|^1/n)


The Attempt at a Solution


a) Is cn in this case just 1? And plugging it in, the radius is 1?

b) cn = n+2[tex]^{n}[/tex], so then limsup n=> infinity |n+2[tex]^{n}[/tex]|[tex]^{1/n}[/tex] => ?? I'm stuck at this point.

i'm also confused in general, is cn just a sequence of coefficients, and what is zn? And I have other formulas for figuring out the radius of convergence, such as the ratio test. I'm not sure when to use which methods. Thank you!
 
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  • #2
I also don't understand why z^n isn't used in the calculation of the radius of convergence.
 
  • #3
So I'm running into trouble for part b still, any help would be greatly appreciated. The answer key says 1/2, but I don't know how to derive that.
 

What is the radius of convergence in complex analysis?

The radius of convergence in complex analysis refers to the distance from the center of a power series where the series converges. It is a measure of how quickly or slowly a series approaches its limit as the degree of the polynomial increases.

How is the radius of convergence determined?

The radius of convergence is determined by the Cauchy-Hadamard theorem, which states that the radius of convergence is equal to the reciprocal of the limit superior of the nth root of the absolute value of the coefficients of the power series.

What happens when the radius of convergence is infinite?

If the radius of convergence is infinite, then the power series converges for all values of the complex variable. This means that the function represented by the power series is analytic on the entire complex plane.

Can the radius of convergence be negative?

No, the radius of convergence cannot be negative. It is always a positive value or can be infinite. A negative radius of convergence would indicate that the power series is divergent, meaning it does not have a well-defined limit.

How is the radius of convergence used in complex analysis?

The radius of convergence is used to determine the domain of convergence for a power series and to analyze the behavior of a function near its singular points. It is also useful in approximating functions and in solving differential equations.

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