The Taylor series as: sum_(n=0)^(oo)f^(n)(a)/(n!)(x−a)^n where a=0 is the Maclaurin series. what is f(a) for the Taylor Series that results in this MacLaurin series? How this is derived?

acsuvaic9

acsuvaic9

Answered question

2022-07-18

The Taylor series as: n = 0 f ( n ) ( a ) n ! ( x a ) n
where a = 0 is the Maclaurin series.
The Maclaurin series for 1 1 x is the geometric series 1 + x + x 2 + . . .
what is f ( a ) for the Taylor Series that results in this MacLaurin series?
then goes into the Taylor series for 1 x at a = 1 is 1 ( x 1 ) + ( x 1 ) 2 ( x 1 ) 3 + . . .
how this is derived?

Answer & Explanation

eyiliweyouc

eyiliweyouc

Beginner2022-07-19Added 15 answers

If we take the function f ( x ) = 1 / ( 1 x ) for | x | < 1, then f ( n ) ( x ) = n ! / ( 1 x ) n for all n [by induction], so f ( n ) ( 0 ) = n ! for all n, so the Maclaurin series is
n = 0 f ( n ) ( 0 ) n ! ( x 0 ) n n = 0 n ! n ! ( x 0 ) n = n = 0 x n .
If we take f ( x ) = 1 x for x > 0, then f ( n ) ( x ) = ( 1 ) n n ! x n + 1 and f ( n ) ( 1 ) = ( 1 ) n . The Taylor series at a = 1 is n = 0 f ( n ) ( 1 ) n ! ( x 1 ) n = n = 0 ( 1 ) n n ! n ! ( x 1 ) n = n = 0 ( 1 ) n ( x 1 ) n .

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