# Sum of inverse of pairwise square roots [closed]

Prove that $$\sum_{i=1}^{50} \frac{1}{\sqrt{2i-1}+\sqrt{2i}} \gt \frac 92$$

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## closed as off-topic by Matthew Conroy, Cameron Buie, Lost1, LTS, J. W. PerryJan 30 at 19:13

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have you tried using: $\frac{1}{\sqrt{2i-1}+\sqrt{2i}}=\sqrt{2i}-\sqrt{2i-1}$ ? –  b00n heT Jan 30 at 11:06
@boon heT I did but that leads to nothing more than an inequality of greater than 3.5 –  Akshit Jan 30 at 16:10
I've tried the integral test, but it ultimately leads to the inconclusive result that $\sum>5-\dfrac1{\sqrt2}$, which however is smaller than $\dfrac92$. –  Lucian Jan 30 at 18:24
Using @b00nheT's comment you get the exact value $10 - 1 = 9 > 9/2$. –  vonbrand Jan 30 at 18:45
@vonbrand: No, they don't cancel so easily! –  TonyK Jan 30 at 18:52