Among many images of carved pumpkin, I like the one above (see Fig. 1) the most. It shows Leibniz’s formula for calculating the value of . Namely,
To derive this formula, we begin with finding the derivative of :
Let , we have , and
It follows that by (1) and the Fundamental Theorem of Calculus,
From carrying out polynomial long division, we observe
It seems that
Assuming (3) is true, we integrate it with respect to from 0 to 1,
As a result of integration, (2) becomes
Moreover, , we obtain through solving . It means that such that for all , i.e.,
The numerical value of is therefore approximated according to (4) by the partial sum
Its value converges to as increases.
However, (5) is by no means a practical way of finding the value of , since its convergence is so slow that many terms must be summed up before a reasonably accurate result emerges (see Fig. 2)
I doubt Leibniz has ever used his own formula to obtain the value of !
Let me leave you with an exercise: Prove (3)