## anonymous 3 years ago What is the expression for exact value of pi - using infinite series?

1. anonymous

Are you allowed to use any series?

2. anonymous

any series will do

3. 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.

4. anonymous

but that's only my first intuition and maybe it doesn't answer that problem rigorously

5. anonymous

$\lim_{x \to \infty} \tan^{-1} x = \frac{\pi}{2}$ $\lim_{x \to \infty} 2 \tan^{-1}x = 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}x^{2n+1}$

6. 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

7. anonymous

$\sum_{n=0}^{+ \infty }\frac{(-1)^n}{2n+1}=\frac{ \pi}{4}$