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Math Help - Question - Power/taylor/MacLaurin series

  1. #1
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    Question - Power/taylor/MacLaurin series

    Hello all; a homework question that is confusing me:

    1) Find the Maclaurin Series for this function: f[x] = e^x + 2e^-x. Can I just make this

    Sum(0 to Infinity) of x^n/(n!) + 2Sum(0 to Infinity) of (-x)^n/(n!) ??
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Sprintz View Post
    Hello all; a homework question that is confusing me:

    1) Find the Maclaurin Series for this function: f[x] = e^x + 2e^-x. Can I just make this

    Sum(0 to Infinity) of x^n/(n!) + 2Sum(0 to Infinity) of (-x)^n/(n!) ??
    Yes, but you can combine some terms:

    \sum_{n=0}^{\infty}\frac{x^n}{n!}+2\sum_{n=0}^{\in  fty}\frac{(-x)^n}{n!} = 1+2+x-2x+\frac{x^2}{2}+2\frac{x^2}{2}+\frac{x^3}{6}-2\frac{x^3}{6}+...= 3-x+\frac{3x^2}{2}-\frac{x^3}{6}+...

    If n is even, the coefficient is 3; if n is odd, -1; it can therefore be represented as 1+2(-1)^n

    So the series is \sum_{n=0}^{\infty}\frac{(1+2(-1)^n)x^n}{n!}
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