Finding the Taylor series of a function

Amaelle
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Homework Statement
look at the image
Relevant Equations
taylor's series
Greetings!
1644257151464.png


Here is the solution that I understand very well I reach a point I think the Professor has mad a mistake , which I need to confirm

after putting x-1=t
we found:

1644257503194.png

But in this line I think there is error of factorization because we still need and (-1)^(n+1) over 3^n

Thank you!
Best regards!
 
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Thank you , I was right!

Best regards!
 
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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