Talk:Leibniz formula for pi
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The justification of the term-by-term integration here is not actually trivial. Charles Matthews 12:34, 9 Mar 2005 (UTC)
[edit] Better proof
I was trying to find π using a power series in one way or another in order to get the Leibniz formula, and I found this to be a better method of showing how:
We know that 
Take the first derivative of arctan(x) and put it in a geometric series:
Integrate that:
So, we have a power series for arctan(x):
Plug in 1 for x and get π/4:
For aesthetics (article uses n=0 to ∞):
Q.E.D.
-Matt 20:11, 18 March 2006 (UTC)
"However, if the series is truncated at the right time, the decimal expansion of the approximation will agree with that of π for many more digits, except for isolated digits or digit groups."
I feel there should be more elaboration on this.
- Hint: if N is a power of ten, each term in the right sum will be a finite decimal fraction. I agree that there is room for elaboration in the article. Feel free to edit it. Fredrik Johansson 11:29, 13 September 2006 (UTC)
