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Topic: [ap-calculus] Re: Convergence of pi
Replies: 1   Last Post: Apr 10, 2002 11:09 PM

 MAXINE BRIDGER Posts: 63 Registered: 12/6/04
[ap-calculus] Re: Convergence of pi
Posted: Apr 10, 2002 11:09 PM

Your curious discovery is not just a coincidence. There is an explanation.
There is an article about it in the MAA journal, The American Mathematical
Monthly. The article appeared in October 1989 and is entitled, "Pi, Euler
Numbers,and Asymptotic Expansions" by Borwein, Borwein and Dilcher.
Actually, my husband Mark remembered where it was and found the reference.

Maxine Bridger

On Tue, 9 Apr 2002, Stu Schwartz wrote:

> Today in BC calculus, one of my students discovered something which I
> guess is a coincidence, But I was curious if anyone ever saw it before.
>
> I was showing them how to develop a power series for arc tan (x).
>
> We created a power series for 1/(1+x) = 1 - x + x^2 - x^3 + x^4 + ...
>
> So f(x^2) = 1/(1+x^2) = 1 - x^2 + x^4 - x^ 6 + x^8 + ...
>
> Integrate each side and get arc tan x = x - x^3/3 + x^5/5 - x^7/7 + ...
>
> So at x = 1, we get pi/4 = 1 - 1^3/3 + 1^5/5 - 1^7/7 + ...
>
> and pi = 4 - 4^3/3 + 4^5/5 - 4^7/7 + ...
>
>
> I wanted to show my students that this very, very slowly converges to
> pi. So I looked at the sum of the first 50 terms on the TI-83.
>
> The sum of the first 50 terms is 3.121594653
> The 9 decimal places of pi is 3.141592653
> Note that they are exactly the same except for the 2nd decimal place -
> 2 vs 4) and the 6th decimal place - 4 vs 2).
>
> These numbers do not stay the same - the 49 term sum and 51st term sum
> are quite different.
>
> Interestingly enough, the 3rd - 5th decimal place numbers (159) also
> crops up in the sum of the 100th term sum.
>
>
> Did this on Excel to 100 terms and found that that the 4th and 5th
> decimal place - 59 also shows up in other sums of terms, usually
> multiples of 50. Terms 200, 250, 300, 500, and 1000.
>
> I can't think this is anything but a coincidence but is certainly a
> strange one.
>
> Anyone ever seen this before?
>
> - Stu Schwartz
> Wissahickon High
>
>
>
>
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