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/sci/ - Science & Math

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8461337 No.8461337 [Reply] [Original]

Is there a closed form for
[eqn]\sum_{x=1}^n \frac{4x + \sqrt{4x^2-1}}{ \sqrt{2x+1} + \sqrt{2x-1} }[/eqn], possibly in terms of the usual special functions?

Wolfram Alpha doesn't seem to think so, but I can't shake off the feeling that there's some sort of linear combination/ hypergeometric function/ convolution trickery to yield a solution.

>> No.8461346

>>8461337
Well, the terms in the sum simplify to (2x+1)^(3/2)-(2x-1)^(3/2)
Which means it turns into a telescopic sum, so only the first and last terms "survive"
S=(2n+1)^(3/2)-1

>> No.8461354
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8461354

>>8461346
Huh, you're right (albeit up to a constant factor of two in the denominator, so the result should be S/2). How did I miss that?

More importantly, does this make Wolfram Alpha infallible?

>> No.8461355

>>8461354
>infallible
*fallible goddammit i need to work on my english in addition to my math

>> No.8461357

>>8461354
wolfra alpha has plenty of errors and problems