# Prove: √(2n) / ∑( 1/√(2n+1) - (1/2n) ) converges to 1 as n -> ∞?

This question is inspired by another one posted by KT:

http://answers.yahoo.com/question/index?...

Of course, the sum is from k = 1 to k = n, and the question is more accurately stated as: Prove that √(2n) / ∑(k = 1 to n) ( 1/√(2k+1) - (1/2k) ) converges to 1 as n -> ∞

http://answers.yahoo.com/question/index?...

Of course, the sum is from k = 1 to k = n, and the question is more accurately stated as: Prove that √(2n) / ∑(k = 1 to n) ( 1/√(2k+1) - (1/2k) ) converges to 1 as n -> ∞

Update:
Zeta, you are correct, the absolute difference increases without bound, which is why I didn't ask for proof that √(2n) is an asymptote. Nonetheless, the ratio does converge to 1.

Update 2:
That was pretty neat, Zo Maar

Update 3:
First Grade Rocks!, a slightly different way is to show that as the n+1 term is added, increasing the sum by 1/√(2n+3) - 1/(2n+2), it approaches the difference √(2n+2) - √(2n) for large n.

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