By looking at an integral and bounding the error?
How closely can we estimate $\sum_{i=0}^n \sqrt{i}$
20
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sequences-and-series
integration
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3What have you *tried* so far? – 2010-09-29
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0Yes I have tried. I got bounds on the sum by but they differ by order of \sqrt{n} which doesn't seem like a great estimate. – 2010-09-29
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0@blaklaybul Compare the sum with $\int_0^n\sqrt{x}dx=2n\sqrt{n}/3$. – 2010-09-30
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0@AD.: I suggest you check out the answers. What you said has already been said in the answers and blaklaybul's comment about sqrt(n) error is exactly about that, I believe! – 2010-10-01
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0@Moron: Sorry, i was a bit hasty (btw it was a nice job you did there). What I wanted to say is that it is easy to get a feel for a sum by looking at a similar integral (which are often much easier to deal with of course). Maybe too trivial comment? – 2010-10-01
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0@AD. Thanks! About the integral, not trivial, but OP already knew that (even before getting any answers, I think), so kind of redundant. – 2010-10-01
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0This sum is investigated at this [MSE link](http://math.stackexchange.com/questions/442470/). – 2015-01-21