Can you help mr to solve that

let c0,c1,c2...in R prove that if lim when n goes to infinity of abs(c(subn)/c(sub n+1)) exists,it is equal to the radius of convergence of the power series c(subn)x^n

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- Nov 14th 2008, 08:18 PMdrpawelseries convergence and radius of convergence
Can you help mr to solve that

let c0,c1,c2...in R prove that if lim when n goes to infinity of abs(c(subn)/c(sub n+1)) exists,it is equal to the radius of convergence of the power series c(subn)x^n - Nov 15th 2008, 09:07 AMMathstud28
Ok let be a sequence in . Ok let us split this up into three cases: , , and .

**Case 1**

Suppose we have a series of the form . Then to determine the radius of convergence I apply the Ratio test. So . So now by our assumption above this is equivalent to , thus the radius of convergence of case 1 is and we are done.

**Case 2**

Now suppose we have the same series but we know have that , which is an identity for . Thus the radius of convergence is .

**Case 3**

Now once again we reach that which only occurs when .

The last two I did more holistically than the first, if you need more rigor I leave that up to you.