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Math Help - representation of functions as power series

  1. #1
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    representation of functions as power series

    a) use differentiation to find a power series representation for f(x)= 1/(1+x)^2
    what is the radius of convergence?
    my answer: sigma from n =1 to infinity of (-1)^n[nx^(n-1)] R = 1 is this right. should n start at one or zero.
    b) use part (a) to find a power series for f(x)= 1/(1+x)^3
    my answer:n =1 to infinity of (-1)^n[n(n-1)x^(n-2)] is this right too R =1
    c) use part (b) to find a power series for f(x)= x^2/(1+x)^3
    my answer: n =1 to infinity of (-1)^n[n(n-1)x^(n)] is this right too R =1
    is this right too?
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    Hello,
    Quote Originally Posted by twilightstr View Post
    a) use differentiation to find a power series representation for f(x)= 1/(1+x)^2
    what is the radius of convergence?
    my answer: sigma from n =1 to infinity of (-1)^n[nx^(n-1)] R = 1 is this right. should n start at one or zero.
    It's correct. It's better to make it start at n=1. Although it's not a big problem to start at n=0, because the n=0 term is 0.

    b) use part (a) to find a power series for f(x)= 1/(1+x)^3
    my answer:n =1 to infinity of (-1)^n[n(n-1)x^(n-2)] is this right too R =1
    Same as above. But following your logic, it should have been n=2 here

    c) use part (b) to find a power series for f(x)= x^2/(1+x)^3
    my answer: n =1 to infinity of (-1)^n[n(n-1)x^(n)] is this right too R =1
    is this right too?
    yes (n=2)
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    however in the answer key (b) and (c) are multiplied by 1/2. why is that?
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    Moo
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    Quote Originally Posted by twilightstr View Post
    however in the answer key (b) and (c) are multiplied by 1/2. why is that?
    Oh no, I'm sorry !! I forgot that one

    When you differentiate \frac{1}{(1+x)^2}=(1+x)^{-2}, what do you get ?
    Recall that the derivative of [u(x)]^n is u'(x) \cdot {\color{red}n} \cdot [u(x)]^{n-1}
    There's also a minus sign that should go with the 1/2
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