pi = 4 - 4/3 + 4/5 - 4/7 + ...

Look at the partial sums (I will use five decimal places because of rounding errors):

4

4 - 4/3 = 8/3 ~ 2.66667

4 - 4/3 + 4/5 = 52/15 ~ 3.46667

4 - 4/3 + 4/5 - 4/7 = 304/105 ~ 2.89524

In each subsequent column below, average the entries that are immediately to the right, and diagonally above to the right,

to get the respective new entry in the next column on the left.

Format:

A

B C

C = (A + B)/2

________________

\(\displaystyle 4 \ \ \ \)

\(\displaystyle 2.66667 \ \ \ \ 3.33332 \)

\(\displaystyle 3.46667 \ \ \ \ 3.06667 \ \ \ \ 3.2\)

\(\displaystyle 2.89524 \ \ \ \ 3.18095 \ \ \ \ 3.12381 \ \ \ \ 3.16191 \)

Now, average the bottom two right entries:

(3.12381 + 3.16191)/2 ~ 3.14 \(\displaystyle \ \ \ \) rounded to two decimal places

Instead of the partial sums, suppose we use the first four terms for \(\displaystyle \ \pi/4\).

a is the 1st term, b is the 2nd term, c is the 3rd term, and d is the 4th term.

There is a short-cut formula which is derived from the procedure with the columns and the partial sums from above:

\(\displaystyle \ \dfrac{1}{16}\bigg[16a + 15b + 10c + 3d\bigg]\)

So, \(\displaystyle \ \dfrac{\pi}{4} \ \approx\ \dfrac{1}{16}\bigg[16(1) + 15(-1/3) + 10(1/5) + 3(-1/7)\bigg] \ \approx \ 0.78571\)

Then, \(\displaystyle \ \pi\ \approx\ \ 3.14\)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

You can use more terms, and if there are enough, you can gain more decimal digits.

For example, \(\displaystyle \ log_e(2) \ \) = 1 - 1/2 + ... - 1/8, out to eight terms.

We can delay the formula and let a = 1/5, b = -1/6, c = 1/7, and d = -1/8.

\(\displaystyle \ log_e(2) \ \approx \ 1 - 1/2 + 1/3 - 1/4 \ + \ \dfrac{1}{16}\bigg[16(1/5) + 15(-1/6) + 10(1/7) + 3(-1/8)\bigg] \ \approx \ 0.693\)