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?

Question

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?

eyiliweyouc · Accepted Answer

If we take the function   f  (  x  )  =  1      /    (  1  −  x  ) for       |    x      |    &amp;lt;  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  &amp;gt;  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    .

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?

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