Leibniz's Formula for Pi/Proof by Taylor Expansion
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Theorem
- $\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$
That is:
- $\ds \pi = 4 \sum_{k \mathop \ge 0} \paren {-1}^k \frac 1 {2 k + 1}$
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.
$\blacksquare$