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
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.