Is there a way to derive the Maclaurin series for 1 ( 1 − x ) after finding the Maclaurin series for ( 1 + x ) n which is ∑ k = 0 ∞ f k ( 0 ) k ! ∗ x k .

Question

Is there a way to derive the Maclaurin series for       1          (      1      −      x      )       after finding the Maclaurin series for   (  1  +  x      )    n   which is             ∑              k        =        0            ∞                                f          k                (        0        )                    k        !              ∗          x      k      .

Larissa Hart · Accepted Answer

Elaborating on J.M. answer, you can procede this way:  (  1  +  x      )    n    =  1  +  n  x  +      1    2    (  n  −  1  )  n      x    2    +      1    6    (  n  −  2  )  (  n  −  1  )  n      x    3    +  …, more generally,  (  1  +  x      )    α    =  1  +      ∑          k      =      1              n                          (                    α      k                      )                  x    k    +  o  (      x    n    )for any real   α, where                     (                    α      k                      )              =            (      α      −      1      )      (      α      −      2      )      …      (      α      −      k      +      1      )              k      !       is the binomial coefficient extended to any real number. Then, you get that  (  1  −  x      )    n    =  1  −  n  x  +      1    2    (  n  −  1  )  n      x    2    −      1    6    (  n  −  2  )  (  n  −  1  )  n      x    3    +  …then, you only have to plug in   n  =  −  1 to get      1          1      −      x        =  1  +  x  +      x    2    +      x    3    +  …

Is there a way to derive the Maclaurin series for 1/(1−x) after finding the Maclaurin series for (1+x)^n which is sum_(n=0)^(oo) f^k(0)/(k!)∗x^k.

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