Euler's formula for pi
The value π = 3,1415… can be calculated with the formula developed by the Swiss mathematician Leonhard Euler
Explanation
You can convert this improper integral to
Evaluating the integrals in the last sum by numerical integration we get
k 0 1.85194 1 .43379 2 .25661 3 .18260 ∆ ∆2 ∆3 ∆4 4 .14180 −2587 5 .11593 799 −1788 −321 6 .09805 478 153 −1310 −168 7 .08495 310 −1000 8 .07495
The sum to k = 3 is
1.85194 − .43379 + .25661 − .18260 = 1.49216
Applying the Euler transform to the remainder we obtain
We obtain the value of the integral as
1.4962 + .07862 = 1.57078
as compared with 1.57080.
HistoryThe Swiss mathematician Leonhard Euler (1707 - 1783) developed this numerical integration. |