Differentiating and Integrating Power Series

Key Questions

  • How do you find the #n#-th derivative of a power series?

    If #f(x)=sum_{k=0}^infty c_kx^k#, then
    #f^{(n)}(x)=sum_{k=n}^infty k(k-1)(k-2)cdots(k-n+1)c_kx^{k-n}#

    By taking the derivative term by term,
    #f'(x)=sum_{k=1}^infty kc_kx^{k-1}#
    #f''(x)=sum_{k=2}^infty k(k-1)c_kx^{k-2}#
    #f'''(x)=sum_{k=3}^infty k(k-1)(k-2)c_kx^{k-3}#
    .
    .
    .
    #f^{(n)}(x)=sum_{k=n}^infty k(k-1)(k-2)cdots(k-n+1)c_kx^{k-n}#

    Wataru · · Sep 12 2014
  • How do you find the antiderivative of a power series?

    If #sum_{n=0}^infty c_n x^n# is a power series, then its general antiderivative is

    #intsum_{n=0}^infty c_n x^n dx=sum_{n=0}^infty c_n/{n+1}x^{n+1}+C#.

    (Note that integration can be done term by term.)

    I hope that this was helpful.

    Wataru · · Oct 19 2014
  • How do you find the derivative of a power series?

    One of the most useful properties of power series is that we can take the derivative term by term. If the power series is

    #f(x)=sum_{n=0}^inftyc_nx^n#,

    then by applying Power Rule to each term,

    #f'(x)=sum_{n=0}^infty c_n nx^{n-1}=sum_{n=1}^inftync_nx^{n-1}#.

    (Note: When #n=0#, the term is zero.)

    I hope that this was helpful.

    Wataru · · Oct 2 2014

Questions

Power Series