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Old 01-16-2012, 01:31 AM   #1
Gigacat
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Default My Maths theory

i was being bored last night so i played around on my calculator trying to find an infinite series that limits at pi. (if your confused at this point, i dont reccomend continuing) Instead, if found this:

(1/n)+(1/n^2)+(1/n^3)+(1/n^4)... = 1/(n-1)

I added the brackets to emphasize the pattern.

I did a bit of googling around to see if i discovered this series and to my surprise, I couldn't find any mention of it on any websites. i find that a bit hard to believe because it seems like such a simple series.

I put the series straight in to google and didn't find anything of interest but if anyone else finds something or if I'm searching wrong, or if they know of the series, let me know.
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Old 01-16-2012, 01:35 AM   #2
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Default Re: My Maths theory

You're absolutely right. What you've found is called a geometric series. It converges whenever |n| > 1. More generally, the series a + ar + ar^2 + ar^3 + ... + ar^n gives you a * (1 - r^n) / (1 - r), and when you let a = 1, let |r| < 1, and let n go to infinity, you get the result you've described.

There aren't all that many nice series that converge to pi. 1/1 - 1/3 + 1/5 - 1/7 + ... converges to pi/4, but everything else I've found in terms of pi involves lots of ugly terms or much more advanced math than I know.
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Old 01-16-2012, 03:00 PM   #3
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Default Re: My Maths theory

It doesn't work with n=99. You've found something, but the first number I put in (99) wasn't adding up. After four sets I only got .0102040815. Ir has pretty set limitations.
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Old 01-16-2012, 03:15 PM   #4
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Default Re: My Maths theory

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Originally Posted by Wackodude View Post
It doesn't work with n=99. You've found something, but the first number I put in (99) wasn't adding up. After four sets I only got .0102040815. Ir has pretty set limitations.
1/(99-1)=0.0102040816 so it actually is correct.

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Old 01-17-2012, 01:33 PM   #5
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Default Re: My Maths theory

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Originally Posted by Wackodude View Post
It doesn't work with n=99. You've found something, but the first number I put in (99) wasn't adding up. After four sets I only got .0102040815. Ir has pretty set limitations.
I could actually prove that Geometric series work via induction, but I don't really feel like crunching the algebra.
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Old 01-26-2012, 11:11 PM   #6
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1/(99-1)=0.0102040816 so it actually is correct.
What ding-dong difference does dat make? None. It's reaching an asymptote, and it won't be pi.
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Old 01-27-2012, 01:38 AM   #7
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Default Re: My Maths theory

I never said that it would. I was playing round trying to find an approximation and i didnt. Instead, i found that.
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Old 01-27-2012, 10:09 PM   #8
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Default Re: My Maths theory

Okay.

What number did you use to compute pi, Gigacat? I think you may have something here. There's a million digits of Pi website, and if you could compare it to that...
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Old 01-27-2012, 11:59 PM   #9
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Default Re: My Maths theory

pi = 4(1 - 1/3 + 1/5 - 1/7 + ...) (From the Taylor series for arctan evaluated at 1).

pi = lim[n->infinity] 2^n * sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + ...) (With n square roots, from the geometry of a circle).

pi = 6 * sqrt(1/1^2 + 1/2^2 + 1/3^2 + ...) (zeta(2), proved by Euler).

There are others, but they get much messier and aren't terribly informative. These let you calculate pi to as many decimals as you like, but the numerical value isn't all that useful from a mathematical perspective. I'd be much more interested in a series that can be nicely compared with e = 1/0! + 1/1! + 1/2! + ... . Or, failing that, a nice geometric interpretation of e.
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Old 01-30-2012, 08:07 PM   #10
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Default Re: My Maths theory

Major kudos to Cheeselord for bringing in the Euler number! I now am surrounded by math geeks. I'm going to go memorize the square root of 13 now, to get at least SOME cred around here...
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Old 01-30-2012, 08:14 PM   #11
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Default Re: My Maths theory

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1/0!
Wouldn't that cause a division by 0 error? AFAIK, 0! is still 0
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Old 01-31-2012, 12:21 AM   #12
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Default Re: My Maths theory

0! is defined to be 1. It makes a lot of branches of mathematics much nicer. You can think of it as the empty product, the number of ways to arrange 0 items, or the reciprocal of the coefficient for the 0th derivative in the Taylor expansion.
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Old 01-31-2012, 01:31 PM   #13
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Default Re: My Maths theory

Quote:
Originally Posted by Wackodude View Post
Major kudos to Cheeselord for bringing in the Euler number! I now am surrounded by math geeks. I'm going to go memorize the square root of 13 now, to get at least SOME cred around here...
A very cool thing is the proof that the squareroot of 13 is not rational. (i.e. that it cannot be written as two integers divded by each other)
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Old 01-31-2012, 06:18 PM   #14
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Default Re: My Maths theory

But, pi and the euler number aren't rational either.
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