Ratio Test for Radius of Convergence | Solving sum(5^n)((x-3)^n)/n"

thomas12323
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Homework Statement



The problem is looking for the radius of convergence
sum(5^n)((x-3)^n)/n
n=1

Homework Equations





The Attempt at a Solution

 
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What work have you done in using the ratio test?
 
I broke it down to (x-3)n/(n+1)
 
That's not right. What happened to the 5^n factor?
You should have this as part of your work.
\lim_{n \to \infty} \left(\frac{5^{n + 1}(x - 3)^{n + 1}}{n + 1}~\frac{n + 1}{5^n (x - 3)^n}\right )
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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