# The value of x

• Apr 1st 2012, 04:10 AM
Mhmh96
The value of x
• Apr 1st 2012, 04:17 AM
Prove It
Re: The value of x
• Apr 1st 2012, 04:24 AM
princeps
Re: The value of x
Quote:

Originally Posted by Mhmh96

$x=\frac{1}{\sqrt 2} \cdot \displaystyle \sum_{n=0}^{1003} \frac{1}{\sqrt n +\sqrt{n+1}}$
• Apr 1st 2012, 04:37 AM
Mhmh96
Re: The value of x
• Apr 1st 2012, 08:01 AM
Wilmer
Re: The value of x
FOR k = 0 to 2006 STEP 2
x = x + 1 / [SQRT(k) + SQRT(k+2)]
NEXT k

x = 22.4053565...

SURPRISE(?): x^2 = 502 (integer) (Wondering)
• Apr 1st 2012, 09:22 PM
browni3141
Re: The value of x
It appears that $\sum\frac{1}{\sqrt{2n}+\sqrt{2(n+1)}}$from n = 0 to k is $\sqrt{(k+1)/2}$
It is almost certain that $x = \sqrt{502}$
• Apr 2nd 2012, 09:16 AM
Wilmer
Re: The value of x
Well, for all the good this'll do:

1 / [SQRT(n) + SQRT(n + 2)] = SQRT{[n + 1 - SQRT(n^2 + 2n)] / 2}
• Apr 9th 2012, 07:43 PM
browni3141
Re: The value of x
I'm surprised no one was able to figure this out until now (including me). It's actually fairly simple.
$x=\frac{1}{\sqrt 2} \cdot \displaystyle \sum_{n=0}^{1003} \frac{1}{\sqrt n +\sqrt{n+1}}$
$x=\frac{1}{\sqrt 2} \cdot \displaystyle \sum_{n=0}^{1003} \frac{\sqrt n -\sqrt{n+1}}{(\sqrt n -\sqrt{n+1})(\sqrt n +\sqrt{n+1})}$
$x=\frac{1}{\sqrt 2} \cdot \displaystyle \sum_{n=0}^{1003} \sqrt{n+1} -\sqrt n$
Now it is a telescoping series.
$x=\frac{1}{\sqrt 2}(\sqrt{1003+1} -\sqrt 0)$
$x= \sqrt{502}$