Madhava's correction term

Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant π (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for π. The Madhava–Leibniz infinite series for π is

${\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }$

Taking the partial sum of the first ${\displaystyle n}$ terms we have the following approximation to π:

${\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}}$

Denoting the Madhava correction term by ${\displaystyle F(n)}$, we have the following better approximation to π:

${\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}+(-1)^{n}F(n)}$

Three different expressions have been attributed to Madhava as possible values of ${\displaystyle F(n)}$, namely,

${\displaystyle F_{1}(n)={\frac {1}{4n}}}$

${\displaystyle F_{2}(n)={\frac {n}{4n^{2}+1}}}$

${\displaystyle F_{3}(n)={\frac {n^{2}+1}{4n^{3}+5n}}}$

In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms ${\displaystyle F_{1}(n)}$ and ${\displaystyle F_{2}(n)}$ have been obtained, but there are no indications on how the expression ${\displaystyle F_{3}(n)}$ has been obtained. This has led to a lot of speculative work on how the formulas might have been derived.