Euler's Estimate of Pi
Euler's Estimate of Pi
In [1] Euler derived the formula . He claimed that his formula was better for calculation than the Leibniz–Gregory formula , since for , the factor in the series has values ,,, which are simpler to calculate with. He illustrated this with the formula . He calculated eight terms of the sum for each of the arc tangents on the right to 27 decimal places each and concluded that . On the next page he calculated terms 9–16 of the first part and terms 9–10 of the second part and concluded that . To eighteen places, . To 30 places, .
arctan(t)=
t
1+
2
t
∞
∑
i=0
(2i)!!
(2i+1)!!
i
2
t
1+
2
t
arctan(t)=
∞
∑
i=0
2i-1
t
2i-1
t=,,
1
3
1
7
3
79
2
t
1+
2
t
1
10
2
100
144
100000
π=8arctan+4arctan
1
3
1
7
π≈3.14159265
π≈3.14159265358979315
π≈3.14159265358979324
π≈3.14159265358979323846264338328