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Deathfish Group Title

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

  • 2 years ago
  • 2 years ago

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  1. Spacelimbus Group Title
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    Are you allowed to use any series?

    • 2 years ago
  2. Deathfish Group Title
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    any series will do

    • 2 years ago
  3. Spacelimbus Group Title
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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.

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

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

    • 2 years ago
  6. Spacelimbus Group Title
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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

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

    • 2 years ago
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