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Thread: sum of a series

  1. #1
    aston
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    sum of a series

    hi, i have a series 1-1/3+1/5-1/7+..., which i think is also called the Leibniz formula, i know that to find the nth term you can can use (-1)^n/(2n+1), but i need help trying to find a formula or a way to find the sum of the series for a certain number of terms i.e. the sum of the first 10 terms, can anyone help?
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  2. #2
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    Krizalid's Avatar
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    Your serie is equal to $\displaystyle \frac\pi4.$

    Consider $\displaystyle \arctan x = \int_0^x {\frac{1}
    {{1 + u^2 }}\,du} .$ So,

    $\displaystyle \arctan x = \int_0^x {\frac{1}
    {{1 + u^2 }}\,du} = \int_0^x {\left\{ {\sum\limits_{k = 0}^\infty {( - 1)^k u^{2k} } } \right\}\,du} .$

    Finally

    $\displaystyle \arctan x = \sum\limits_{k = 0}^\infty {( - 1)^k \left\{ {\int_0^x {u^{2k} \,du} } \right\}} = \sum\limits_{k = 0}^\infty {\frac{{( - 1)^k x^{2k + 1} }}
    {{2k + 1}}} ,\,\left| x \right| \le 1.$

    Now set $\displaystyle x=1$ and we're done.
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    Your serie is equal to $\displaystyle \frac\pi4.$

    Consider $\displaystyle \arctan x = \int_0^x {\frac{1}
    {{1 + u^2 }}\,du} .$ So,

    $\displaystyle \arctan x = \int_0^x {\frac{1}
    {{1 + u^2 }}\,du} = \int_0^x {\left\{ {\sum\limits_{k = 0}^\infty {( - 1)^k u^{2k} } } \right\}\,du} .$

    Finally

    $\displaystyle \arctan x = \sum\limits_{k = 0}^\infty {( - 1)^k \left\{ {\int_0^x {u^{2k} \,du} } \right\}} = \sum\limits_{k = 0}^\infty {\frac{{( - 1)^k x^{2k + 1} }}
    {{2k + 1}}} ,\,\left| x \right| \le 1.$

    Now set $\displaystyle x=1$ and we're done.
    You cannot do that. Because $\displaystyle \frac{1}{1+u^2} = \sum_{k\geq 0} (-1)^k u^{2k}$ only when $\displaystyle |u|<1$. What you are doing is letting $\displaystyle u=1$ and it does not converge there.
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  4. #4
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    Quote Originally Posted by aston View Post
    hi, i have a series 1-1/3+1/5-1/7+..., which i think is also called the Leibniz formula, i know that to find the nth term you can can use (-1)^n/(2n+1), but i need help trying to find a formula or a way to find the sum of the series for a certain number of terms i.e. the sum of the first 10 terms, can anyone help?
    Proof 1: First note that $\displaystyle \frac1{1+x^2} = 1 - x^2 + x^4 -...$ for $\displaystyle |x|<1$. This tells us that (integrate both sides) $\displaystyle \tan^{-1} x = x - \frac{x^3}{3}+\frac{x^5}{5}-...$ for $\displaystyle |x|<1$. Now to justify what Krizalid did is to use Abel's Theorem which tells us that the series $\displaystyle f(x)$ is continous at $\displaystyle x=1$ because it converges there (by alternating series test). Thus, if $\displaystyle x_k\to 1$ is a sequence in $\displaystyle (-1,1)$ then $\displaystyle \lim ~ f(x_k) = f(\lim x_k)$ thus $\displaystyle f(1) = \lim ~ f(x_k) = \lim ~ \tan^{-1} (x_k) = \frac{\pi}{4}$.

    Abel's Theorem: If $\displaystyle f(x) = \sum_{n=0}^{\infty}a_nx^n$ is a power series of radius of convergence of radius $\displaystyle 0<R<\infty$ such that the series converges at $\displaystyle x=R$ then $\displaystyle f$ is continous at $\displaystyle x=R$.
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  5. #5
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    Here is a more elementary proof.

    Proof 2: For any $\displaystyle x$ we have $\displaystyle \sum_{k=0}^{n}(-1)^k x^{2k} = \frac{1 +(-1)^{n+1} x^{2n+1}}{1+x^2}$. Thus, $\displaystyle \int_0^1 \sum_{k=0}^n (-1)^k x^{2k} dx = \int_0^1 \frac{dx}{x^2+1} \pm \int_0^1 \frac{x^{2n+1}}{x^2+1} dx$ where $\displaystyle \pm $ depends on the parity of $\displaystyle n$. This tells us that $\displaystyle \sum_{k=0}^n \frac{(-1)^k x^{2k+1}}{2k+1} = \tan^{-1} 1 \pm \int_0^1 \frac{x^{2n+1}}{x^2+1} dx$. Thus, $\displaystyle \left| \sum_{k=0}^n \frac{(-1)^k x^{2k+1}}{2k+1} - \frac{\pi}{4}\right| = \int_0^1 \frac{x^{2n+1}}{x^2+1} dx$. Now $\displaystyle \frac{x^{2n+1}}{x^2+1} \leq x^{2n+1}$. This means, $\displaystyle \left| \sum_{k=0}^n \frac{(-1)^k x^{2k+1}}{2k+1} - \frac{\pi}{4}\right| \leq \int_0^1 x^{2n+1} dx = \frac{1}{2n+2} \leq \frac{1}{n}$. This means for any $\displaystyle \epsilon > 0$ if we chose $\displaystyle N> \frac{1}{\epsilon}$ it will mean that $\displaystyle \sum_{k=0}^N \frac{(-1)^kx^{2k+1}}{2k+1}$ is within $\displaystyle \epsilon$ of $\displaystyle \frac{\pi}{4}$. And so, by definition, this series converges to $\displaystyle \frac{\pi}{4}$.
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