Question

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

Answer 1

Are you allowed to use any series?

Answer 2

any series will do

Answer 3

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.

Answer 4

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

Answer 5

\[ \lim_{x \to \infty} \tan^{-1} x = \frac{\pi}{2} \]

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

Answer 6

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

Answer 7

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