anonymous
  • anonymous
What is the expression for exact value of pi - using infinite series?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
Are you allowed to use any series?
anonymous
  • anonymous
any series will do
anonymous
  • anonymous
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.

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anonymous
  • anonymous
but that's only my first intuition and maybe it doesn't answer that problem rigorously
anonymous
  • anonymous
\[ \lim_{x \to \infty} \tan^{-1} x = \frac{\pi}{2} \] \[ \lim_{x \to \infty} 2 \tan^{-1}x = 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}x^{2n+1}\]
anonymous
  • anonymous
\[ \large \lim_{x \to \infty} 2 \tan^{-1}x =\lim_{x \to \infty}\ 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}x^{2n+1} \] Shouldn't forget to carry out the limit
anonymous
  • anonymous
\[\sum_{n=0}^{+ \infty }\frac{(-1)^n}{2n+1}=\frac{ \pi}{4}\]

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