Radius of Converge Complex Analysis

[gbean]

Homework Statement

Find the radius of convergence of $$\sum$$cnz$$^{n}$$ if c2k = 2$$^{k}$$ and c2k-1 = (1+1/k)$$^{k^{2}}$$, k = 1, 2, 3...

Homework Equations

1/R = limsup as n=> infinity |cn|^1/n

The Attempt at a Solution

I'm not really sure where to start with this. I know that it's a power series, and to find the radius of convergence, I can use the formula as stated above, but that doesn't seem helpful.

If I tried that, then limsup |c2k|^1/n = (2^k)^(1/n) = 1?

And then, limsup |c2k-1|^1/n = ((1+1/k)^(k^2))^(1/n) = 1? So the radius of convergence of any power series is unique, and the smaller of the 2 radii has to be used, so it's just 1? Is this correct?

Working out the terms of the series, it seems like c2k is adding up to infinity, and c2k-1 is getting infinitely smaller, but never reaches 0. I'm not sure what I'm supposed to derive from this.

 

[mighty2000]

I would approach this problem by first understanding the definition of the radius of convergence and how it relates to the terms of the power series. The radius of convergence is the distance from the center of the power series at which the series will converge. In other words, it is the maximum value of x for which the series will converge.

Next, I would look at the given values for c2k and c2k-1 and try to determine a pattern or relationship between them. I notice that c2k is increasing exponentially while c2k-1 is decreasing exponentially. This suggests that the terms of the series are alternating and may not converge for all values of x.

Then, I would apply the formula for the radius of convergence, |cn|^1/n, to the given values of c2k and c2k-1. This would give me limsup |c2k|^1/n = 2 and limsup |c2k-1|^1/n = 1. This means that the radius of convergence is 1/2, since it is the smaller of the two values.

Finally, I would test the convergence of the series at x = 1/2 and x = -1/2 to confirm that the series does indeed converge within this radius. This can be done by using the ratio test or the root test.

In conclusion, the radius of convergence for the given power series is 1/2. This means that the series will converge for all values of x within this radius, and may or may not converge for values outside of this radius.