Is it always valid to say that the Taylor series of some function f ( x ) about a point x = a equivalent to the the Maclaurin series of another function h ( x ) = f ( x + a )?

Question

Is it always valid to say that the Taylor series of some function   f  (  x  ) about a point   x  =  a equivalent to the the Maclaurin series of another function   h  (  x  )  =  f  (  x  +  a  )?

reinmelk3iu · Accepted Answer

No. If we consider an infinitely differentiable function   f  (  x  ), then we can write down it&#039;s Taylor series about the point   x  =  a as      ∑          n      =      0              ∞                          f                  (          n          )                    (      a      )              n      !        (  x  −  a      )    n    .Now, if we consider the translation  h  (  x  )  =  f  (  x  +  a  ) and compute its Maclaurin series, we get      ∑          n      =      0              ∞                          h                  (          n          )                    (      0      )              n      !            x    n    =      ∑          n      =      0              ∞                          f                  (          n          )                    (      a      )              n      !            x    n    .Note that      ∑          n      =      0              ∞                          f                  (          n          )                    (      a      )              n      !        (  x  −  a      )    n    ≠      ∑          n      =      0              ∞                          f                  (          n          )                    (      a      )              n      !            x    n  because they clearly disagree at   x  =  a. What we can say, however, is that the Taylor series for   f  (  x  ) is the Maclaurin series for   g  (  x  )  =  f  (  x  +  a  ) shifted to the right by   a.

Is it always valid to say that the Taylor series of some function f(x) about a point x=a equivalent to the the Maclaurin series of another function h(x)=f(x+a)?

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