Ratio Test for Finding the 1(circled part) - Understanding the Process

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Homework Statement


where does the 1(circled part) come from ? is it a mistake ?

Homework Equations

The Attempt at a Solution

 

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Try look closer. There is a difference in the second expression apart from the 1.
 

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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