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Deathfish

  • 3 years ago

What is the expression for exact value of pi - using infinite series?

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  1. Spacelimbus
    • 3 years ago
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    Are you allowed to use any series?

  2. Deathfish
    • 3 years ago
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    any series will do

  3. Spacelimbus
    • 3 years ago
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    then I recommend using the series for tan^(-1)x, because I know that this series approaches for very large x the value pi / 2, so you can just take a multiply (2 times) of that.

  4. Spacelimbus
    • 3 years ago
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    but that's only my first intuition and maybe it doesn't answer that problem rigorously

  5. Spacelimbus
    • 3 years ago
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    \[ \lim_{x \to \infty} \tan^{-1} x = \frac{\pi}{2} \] \[ \lim_{x \to \infty} 2 \tan^{-1}x = 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}x^{2n+1}\]

  6. Spacelimbus
    • 3 years ago
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    \[ \large \lim_{x \to \infty} 2 \tan^{-1}x =\lim_{x \to \infty}\ 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}x^{2n+1} \] Shouldn't forget to carry out the limit

  7. Neemo
    • 3 years ago
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    \[\sum_{n=0}^{+ \infty }\frac{(-1)^n}{2n+1}=\frac{ \pi}{4}\]

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