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Math Help - power series

  1. #1
    sprinks
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    power series

    Use series you know to show that:

    1 - 1/3 + 1/5 - 1/7 + ... = pi/4

    I get that arctanx = x - 1/3x^3 + 1/5x^5 - 1/7x^7
    and arctan 1 = pi/4

    but I'm not sure how I'm supposed to show this; what is the connection? Is this the connection I am looking for?
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  2. #2
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    Quote Originally Posted by sprinks View Post
    Use series you know to show that:

    1 - 1/3 + 1/5 - 1/7 + ... = pi/4

    I get that arctanx = x - 1/3x^3 + 1/5x^5 - 1/7x^7
    and arctan 1 = pi/4

    but I'm not sure how I'm supposed to show this; what is the connection? Is this the connection I am looking for?
    Quote Originally Posted by sprinks View Post
    Use series you know to show that:

    1 - 1/3 + 1/5 - 1/7 + ... = pi/4

    I get that arctanx = x - 1/3x^3 + 1/5x^5 - 1/7x^7
    and arctan 1 = pi/4

    but I'm not sure how I'm supposed to show this; what is the connection? Is this the connection I am looking for?
    \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + ... for |x|<1.

    Integrate from 0 to t where |t|<1 we get:
    \int_0^t \frac{dx}{1+x^2} = \int_0^t 1 - x^2 + x^4 - ... dx
    Because of uniform convergence we can integrate term-by-term.
    Thus,
    \tan^{-1} t = t - \frac{t^3}{3} + \frac{t^5}{5} - ... for |t|<1.
    Now this series converges for t=1 so by applying "Abel's theorem" we can evalute this even at t=1 so we get:
    \frac{\pi}{4} = 1 - \frac{1}{3}+ ...
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