My Math Forum Pi formula

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 April 1st, 2010, 07:51 AM #21 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Pi formula Hi, I'm sorry for my late response.. I hope it's the correct term, but for the denominator I find 16k^2+16k+3, so $\pi=\sum_{k=0}^{n} \frac{8}{16k^2+16k+3}$ if initially is correct Hoempa
 April 2nd, 2010, 03:37 AM #22 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Pi formula Has any of you guys ever tried this formula? $\lim_{x\rightarrow\infty}2x\cdot \frac{\sin(\frac{180}{x})}{3} + x\cdot \frac{\tan(\frac{180}{x})}{3}=\pi$ for use: calculate in degrees instead of radian. Approaches pi more accurate. Hoempa
 April 16th, 2010, 02:25 PM #23 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 In your summation, you would need to replace "n" with "?" for equality. In your limit, how would you evaluate the trigonometric functions without already knowing the value of pi?
April 18th, 2010, 09:57 PM   #24
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Re: Pi formula

Hello skipjack,

corrected summation: $\pi=\sum_{k=0}^{\infty} \frac{8}{16k^2+16k+3}$

Quote:
 Originally Posted by skipjack In your limit, how would you evaluate the trigonometric functions without already knowing the value of pi?
Good question, to find it, indeed I used the value of pi. To evaluate, use

$\lim_{x\rightarrow\infty}f(x)= x\cdot \tan(\frac{180}{x})=\pi$
$\lim_{x\rightarrow\infty} g(x)=2x\cdot \frac{\sin(\frac{180}{x})}{3} + x\cdot \frac{\tan(\frac{180}{x})}{3}=\pi$

Compare for example f(10000) and values for g(x), and compare these values, the higher you choose x, the more accurate you approach pi. g(x) converges much faster to pi.

Hoempa

 April 26th, 2010, 02:46 PM #25 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 How accurate a value for pi are you using to evaluate tan(180/x) for, say, x = 10000?
April 27th, 2010, 05:55 AM   #26
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Re: Pi formula

?
Quote:
 Originally Posted by skipjack tan(180/x)
Where did I use this formula?

Hoempa

 May 7th, 2010, 07:19 AM #27 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 In your post dated April 2.
 May 8th, 2010, 08:32 AM #28 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Pi formula There, I used $x \cdot \tan \frac{\frac{180}{x}}{3}$, the formula you mentioned, multiplied with $\frac{x}{3}$. The formula you posted doesn´t seem quite accurate at x=10.000 Hoempa
 May 11th, 2010, 01:41 PM #29 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Pi formula Anyway, I see what you mean. With the value for $\pi$ i'm using from a TI calculator, x=10000 gives in the formula I posted, g(x), $\pi \approx 3.14159265359$, substract pi gives me 0. for f(x), f(10000) on my calculator gives approx 3.14159260191, an error of $-5.16771\cdot 10^{-8}$. Makes me think that g(x) approaches $\pi$ faster then f(x). I have found g(x) by comparing the errors for approximating pi according to my calculator of $x\sin(\frac{180}{x})$ and $x\tan{180}{x}$. edit: I have written a formula that would tell for x, how many decimals you could find from \pi. I'll see if I can find that. Hoempa
 May 13th, 2010, 04:13 AM #30 Senior Member   Joined: Apr 2010 Posts: 215 Thanks: 0 Re: Pi formula In order to calculate this, you will need to know pi, since: $180 ^{\circ}= \pi$ Then you get (for example): $\lim_{x\to\infty} xsin(\frac{\pi}{x})= \pi$ And that would hold true for any number, not just for pi. Thus, if you use an approximation for pi, your result will not be more accurate than your approximation. Or what is it that I don't understand?

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