Leibniz formula for $π$

In mathematics, the Leibniz formula for $π$ , named after Gottfried Wilhelm Leibniz, states that

{\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}},

an alternating series.

It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), and was later independently rediscovered by James Gregory in 1671 and Leibniz in 1673. The Taylor series for the inverse tangent function, often called Gregory's series, is

\arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{2k+1}}.

The Leibniz formula is the special case ${\textstyle \arctan 1={\tfrac {1}{4}}\pi .}$

It also is the Dirichlet $L$ -series of the non-principal Dirichlet character of modulus 4 evaluated at $s=1,$ and therefore the value $β (1)$ of the Dirichlet beta function.

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Leibniz formula for π

Leibniz formula for $π$