1+(1/4) + (1*3)/(4*8) + (1*3*5)/(4*8*12)+(1*3*5*7)/(4*8*12*16)+.......

=sqrt(2)

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- June 10th 2010, 10:08 AMbgbgbgbgshow that the sum = sqrt(2)
1+(1/4) + (1*3)/(4*8) + (1*3*5)/(4*8*12)+(1*3*5*7)/(4*8*12*16)+.......

=sqrt(2) - June 10th 2010, 10:57 AMBruno J.
Nevermind! Listen to tonio. :)

- June 10th 2010, 11:27 AMtonio

That's the sum . Let's try now to understand the quotient in that infinite series a little better:

. There's a reason why we wrote the last mess:

We know (hopefully!) that , with convergence radius . Well, derivate both sides of this and get:

, and now input , do a little algebra and get your result.

Tonio

Ps. This is, imho, a problem that fits the Challenge Problems Section! - June 10th 2010, 03:30 PMawkward
Another way to view the series is that it is the expansion by the Binomial Theorem of

.