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        riddles >> medium >> Summation
        
(Message started by: ThudanBlunder on Apr 23^rd, 2010, 2:52pm)

Title: Summation
Post by ThudanBlunder on Apr 23^rd, 2010, 2:52pm

Find (without programming) the greatest integer less than 1 + 1/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 + 1/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 + ....... + 1/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif1000000

Title: Re: Summation
Post by Obob on Apr 23^rd, 2010, 3:12pm

Given some calculus, it's not so hard to guess the right answer. Proving it is correct (without actually finding the sum) could be much more difficult, though...

Title: Re: Summation
Post by towr on Apr 23^rd, 2010, 3:14pm

[hide]Through integration you can find an upper limit of 1999, and a lower limit slightly over 1998, so 1998 must be the greatest integer that's less[/hide]

Title: Re: Summation
Post by Obob on Apr 23^rd, 2010, 3:18pm

Ah, good call on the upper bound.

Title: Re: Summation
Post by ThudanBlunder on Apr 23^rd, 2010, 3:40pm

Actually, without calculus the sum telescopes nicely if successive terms are 'inverted' somehow. :-X

Title: Re: Summation
Post by Aryabhatta on Apr 23^rd, 2010, 7:45pm

For a without calculus solution, use the following:

[hide]
1/sqrt(n) >= 2(sqrt(n+1) - sqrt(n)) >= 1/sqrt(n+1)
[/hide]