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Math Help - sum of infinite series of functions

  1. #1
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    sum of infinite series of functions

    Find the function, f(x), which the infinite series
    1-(x^3)+(x^6)-(x^9)+(x^12)-(x^15)+....... converges to.
    The hint is "What is 1-x+(x^2)-(x^3)+(x^4)-(x^5)+....."

    I know that 1-x+(x^2)-(x^3)+(x^4)-(x^5)+.... sums to 1/(1+x) for lxl<1 because it is the derivative of the taylor series for ln(1+x) but all I can think of to use that hint is that f(x)=(1/(1+x))+x-(x^2)-(x^4)+(x^5)+(x^7)-(x^8)-(x^10)+(x^11)+.... which doesn't help me much (as far as I can tell).
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by 234578 View Post
    Find the function, f(x), which the infinite series
    1-(x^3)+(x^6)-(x^9)+(x^12)-(x^15)+....... converges to.
    The hint is "What is 1-x+(x^2)-(x^3)+(x^4)-(x^5)+....."

    I know that 1-x+(x^2)-(x^3)+(x^4)-(x^5)+.... sums to 1/(1+x) for lxl<1 because it is the derivative of the taylor series for ln(1+x) but all I can think of to use that hint is that f(x)=(1/(1+x))+x-(x^2)-(x^4)+(x^5)+(x^7)-(x^8)-(x^10)+(x^11)+.... which doesn't help me much (as far as I can tell).
    Since 1-x+x^2-x^3+x^4-x^5+\ldots=\frac{1}{1+x}, take note that

    1-x^3+x^6-x^9+x^{12}-x^{15}+\ldots=1-(x^3)+(x^3)^2-(x^3)^3+(x^3)^4-(x^3)^5+\ldots=\ldots

    I leave it for you to finish this problem.
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    Since 1-x+x^2-x^3+x^4-x^5+\ldots=\frac{1}{1+x}, take note that

    1-x^3+x^6-x^9+x^{12}-x^{15}+\ldots=1-(x^3)+(x^3)^2-(x^3)^3+(x^3)^4-(x^3)^5+\ldots=\ldots

    I leave it for you to finish this problem.


    ohhhhh!! thank you very much. I think I need more sleep, I should have seen that.
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