A while back, we deemed that it is utterly impractical to calculate the value of using the partial sum of Leibniz’s series due to its slow convergence (see “Pumpkin Pi“)

Fig. 1

As illustrated in Fig. 1, in order to determine each additional correct digit of , the number of terms in the summation must increase by a factor of 10.

What we need is a fast converging series whose partial sum yields given number of correct digits with far fewer terms.

Looking back, we see that the origin of Leibniz’s series is the definite integral

To find the needed new series, we consider a variation of (1), namely,

Given the fact (see “Pumpkin Pi“) that

We proceed to integrate (3) with respect to from to ,

As result, (2) becomes

i.e.,

By (4),

which gives

.

And so

Since , (5) implies

.

Hence,

.

It follows that the value of can be approximated by the partial sum of a new series

Let’s compute it with Omega CAS Explorer (see Fig. 2, 3)

Fig. 2

Fig. 2 shows the series converges quickly. The sum of the first 10 terms yields the first 6 digits!

Fig. 3

Totaling the first 100 terms of the series gives the first 49 digits of (see Fig. 3)

Exercise 1. Show that .

Exercise 2. Can we use to compute the value of in a similar fashion? Explain.

Xiaolan XueWow, impressive!