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In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that = + + = = +, an alternating series.
Leibniz discovered his formula for $\pi$ in $1673$. He took great pleasure and pride in this discovery. It's as if, by this expansion, the veil which hung over that strange number had been drawn aside.
Leibniz' formula for is a special case of the Gregory's series for the arctangent, arctan x = x − x 3 3 + x 5 5 − x 7 7 + ⋯ {\displaystyle \arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots } ,
def term(n): return ( (-1.)**n / (2.*n + 1.) )*4. def pi(nterms): return sum(map(term,range(nterms))) and then calculate pi with the number of terms you need to reach a given precision: pi(100) # 3.13159290356 pi(1000) # 3.14059265384
There are many formulas of pi of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. pi is intimately related to the properties of circles and spheres. For a circle of radius r, the circumference and area are given by C = 2pir (1) A = pir^2.
I found the following proof online for Leibniz's formula for $\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute $y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\ldots$$ Now plug in $1$ for $x$: $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots$$
Python Program to Calculate Value of PI Using Leibniz Formula. The Leibniz formula is an infinite series method of calculating Pi. The formula is a very simple way of calculating Pi, however, it takes a large amount of iterations to produce a low precision value of Pi. Leibniz formula: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
The arctan series was obtained by Leibniz and Gregory early in their study of infinite series and, in fact, before the methods and algorithms of calculus were fully developed.
Leibniz’s Formula: Below I’ll derive the series expansion arctan(x) = X∞ n=0 (−1)n x2n+1 2n+1; 0 ≤ x ≤ 1. (1) Plugging the equation π = 4arctan(1) into Equation 1 gives Leibniz’s famous formula for π, namely π = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 ··· (2) This series has a special beauty, but it is terrible for actually ...
$\ds \pi = 4 \sum_{k \mathop \ge 0} \paren {-1}^k \frac 1 {2 k + 1}$ Proof The area $OAT$ is a quarter- circle whose area is $\dfrac \pi 4$ by Area of Circle .