Question

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.

Answer 1

`(1)/(x-1) = \sum\limits_(k=0)^(\infty) (1)/(25^(k+1)) (24-x)^k`

Explanation:

First remember that

`(1)/(1-x) = \sum\limits_(k=0)^(\infty) x^k\\`

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

So we say

`(1)/(x-1) = (1)/(1+x+24-24) = (1)/(25-(24-x)) = (1)/(25(1-((24-x))/(25)))`

`= (1)/(25) \sum\limits_(k=0)^(\infty) ((24-x)/(25))^k = \sum\limits_(k=0)^(\infty) (1)/(25^(k+1)) (24-x)^k`