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Homework Help: Maclaurin series coefficient

  1. Apr 16, 2009 #1
    1. The problem statement, all variables and given/known data
    find coefficient of x^4 in the MAclaurin series for f(x)=e^sinx

    2. Relevant equations
    ok... so taking derivatives 4 times for this function.....gave me a mess!!! @.@
    can someone help me in simplying the derivatives...?
    1. cosxe^sinx
    then for 2. is it -sinxe^sinx-sinxcosxe^sinx....?
    = . = i'm getting lost with these derivatives.....

    same for e^3xcos2x... gets sooo complicated...
    Last edited: Apr 16, 2009
  2. jcsd
  3. Apr 16, 2009 #2
    Don't try to find it by determining the derivatives. You already know the expansions of the function sin x and e^y. Just plug them in, and expand up till 4th order (or better to just look what terms contribute to the x^4 coefficient).

    [tex]e^{\sin x} = \sum_{k=0}^\infty \frac{(\sin x)^k}{k!}[/tex]

    Then plug in the expansion for sin x and collect the terms.
  4. Apr 17, 2009 #3
    hmm i'm still not too sure...
    I do get the series u've written above... now for sinx i expand it like x - x^3/3! + x^5/5!... up to the fourth order.... and plug in 0??.... doesn't that just make everything equal to 0.....????
  5. Apr 18, 2009 #4


    Staff: Mentor

    I think what xepma is talking about is this[tex]e^{\sin x} = \sum_{k=0}^\infty \frac{(\sin x)^k}{k!}[/tex]
    [tex]= 1 + sin(x)/1 + sin^2(x)/2 + sin^3(x)/3! + ...[/tex]

    Now, put in your series for sin(x), sin2(x) and so on. You probably won't need the sin3(x) and might need only a term or two for the sin2(x) part.
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