Thread: Investigate Equality

1. 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.

2. 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?

4. Re: Investigate Equality

My thorough investigation revealed that this is Vičte's formula. Now what?

5. Re: Investigate Equality

Thanks a lot Vlasev, you are truly a good n efficient investigator...good to know the origin of the problem.

6. 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.

Some algebraic manipulation of this will give the result you are trying to prove.

7. Re: Investigate Equality

This explanation is truly enlightening. Thanks a lot ProveIt.

8. Re: Investigate Equality

That's indeed a magnificent proof!