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February 8th, 2018, 09:02 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 104 Thanks: 2  Ratio test complex numbers question:
calculate the radius of convergence for the series: g(z)=\[\sum_{n=1}^{\infty}\frac{z^n}{n^2}\] i used the ratio test to show that An+1/an converges to absolute value of z as n tends to infinity, does this mean that our radius of convergence lies between R=[0,1)? I know that if the limit is equal to 1 for An+1/an then we still do not know anything about the convergence/divergence of the series so i am slightly confused. 
February 8th, 2018, 11:09 AM  #2  
Senior Member Joined: Sep 2015 From: USA Posts: 2,554 Thanks: 1403  Quote:
Mathematica returns values for $polylog(2,z)$ for $z \geq 1$ $polylog(2,1) = \dfrac{\pi^2}{6}$ is pretty well known. A discussion at math stack exchange seems to settle on $z < 1$ so I'm not really sure what the correct answer is. You've opened up a nice can of worms!  
February 8th, 2018, 12:33 PM  #3  
Senior Member Joined: Sep 2016 From: USA Posts: 648 Thanks: 412 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
For example, $f(z) = \frac{1}{z^2 + 1}$ is meromorphic i.e. analytic away from $\pm i$. However, any expansion will have a finite radius of convergence. Namely, if you expand about $z = z_0$, then the radius of convergence will be exactly given by \[ R = \min \{z_0  i, z_0 + i\} \] as this is the distance to the nearest pole. This doesn't mean recentering at another point (which is exactly what analytic continuation is here) doesn't define an extension of the function to other regions of the complex plane.  
February 8th, 2018, 01:03 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 2,554 Thanks: 1403  Quote:
 
February 8th, 2018, 02:53 PM  #5 
Senior Member Joined: Sep 2016 From: USA Posts: 648 Thanks: 412 Math Focus: Dynamical systems, analytic function theory, numerics  
February 9th, 2018, 01:39 AM  #6 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 104 Thanks: 2  How do u know that the series also converges on z=1? Did u just sub 1 into the original series to find the series=1/n^2 which is convergent?


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