Fig. 1

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

.

Since ,

It follows that by (1) and the Fundamental Theorem of Calculus,

i.e.,

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

,

or,

.

Therefore,

.

Moreover, , we obtain through solving . It means that such that for all , i.e.,

.

Thus

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)

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)

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