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Consider the Taylor polynomial T n ( x ) centered at x = 24 for all n for the function f ( x ) = 1 x − 1 , where i is the index of summation. Find the i th term of T n ( x ) . (Express numbers in exact form. Use symbolic notation and fractions where needed. For alternating series, include a factor of the form ( − 1 ) n in your answer.

Respuesta :

Answer:

[tex]\frac{1}{x-1} = \sum\limits_{k=0}^{\infty} \frac{1}{25^{k+1}} (24-x)^k[/tex]

Step-by-step explanation:

First remember that

        [tex]\frac{1}{1-x} = \sum\limits_{k=0}^{\infty} x^k\\[/tex]

We want to manipulate that sum in order for that to be centered around x=24.

So we  say

[tex]\frac{1}{x-1} = \frac{1}{1+x+24-24} = \frac{1}{25-(24-x)} = \frac{1}{25(1-\frac{(24-x)}{25})}[/tex]

[tex]= \frac{1}{25} \sum\limits_{k=0}^{\infty} (\frac{24-x}{25})^k = \sum\limits_{k=0}^{\infty} \frac{1}{25^{k+1}} (24-x)^k[/tex]

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