Leibniz's Formula for Pi/Proof by Taylor Expansion

From ProofWiki
Jump to: navigation, search

Theorem

$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$

This sequence is A003881 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


That is:

$\displaystyle \pi = 4 \sum_{k \mathop \ge 0} \left({-1}\right)^k \frac 1 {2 k + 1}$


Proof

From Power Series Expansion for Real Arctangent Function, we obtain:

$\arctan x = x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \dfrac {x^9} 9 - \cdots$

Substituting $x = 1$ gives the required result.

$\blacksquare$