Investigate Equality

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August 25th 2012, 10:32 PM
MaxJasper

Investigate Equality

Problem: Investigate following equality:

$\frac{2}{\pi }=\frac{\sqrt{2}}{2} \frac{\sqrt{2+\sqrt{2}}}{2} \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \text{...}$

Thanks.
August 25th 2012, 11:39 PM
Vlasev

Re: Investigate Equality

What is there to investigate? Do you want to prove the equality? Do you want to formulate a partial product and see if it converges?
August 26th 2012, 09:14 AM
MaxJasper

Re: Investigate Equality

http://t1.gstatic.com/images?q=tbn:A...gUGkNWZf0Me0lQ
August 26th 2012, 08:08 PM
Vlasev

Re: Investigate Equality

My thorough investigation revealed that this is Viète's formula. Now what?
August 26th 2012, 08:24 PM
MaxJasper

Re: Investigate Equality

Thanks a lot Vlasev, you are truly a good n efficient investigator...good to know the origin of the problem.(Flower)
August 26th 2012, 08:55 PM
Prove It

Re: Investigate Equality

You can define a circle to be a regular polygon with an infinite number of sides. We can define $\displaystyle \begin{align*} \pi \end{align*}$ as the circumference of a circle of diameter equal to 1 unit. Therefore, we can get an approximation for $\displaystyle \begin{align*} \pi \end{align*}$ by evaluating the perimeter of said regular polygon.

http://upload.wikimedia.org/wikipedi...rolled-720.gif

http://i22.photobucket.com/albums/b3...sson7part2.jpg
http://i22.photobucket.com/albums/b3...dragon/pi1.jpg
http://i22.photobucket.com/albums/b3...sson7part3.jpg
http://i22.photobucket.com/albums/b3...dragon/pi2.jpg
http://i22.photobucket.com/albums/b3...sson7part4.jpg
http://i22.photobucket.com/albums/b3...dragon/pi3.jpg
http://i22.photobucket.com/albums/b3...sson7part5.jpg

Some algebraic manipulation of this will give the result you are trying to prove.
August 26th 2012, 09:23 PM
MaxJasper

Re: Investigate Equality

This explanation is truly enlightening. Thanks a lot ProveIt.
August 26th 2012, 10:42 PM
Vlasev

Re: Investigate Equality

That's indeed a magnificent proof!