Saturday, 24 March 2018

calculus - Limit of some specific "almost Riemann" sums



I'm just beginning my journey with calculus, and this problem is giving me hard time.




limnnk=1kn2+k



I calculated first three sums: 12,1630,117220
From that I guessed that the limit is 12. The hard part, though, is proving it.



I reckon that the squeeze theorem should be used here, but I can't seem to find anything useful. (The only sequences I was able to figure out give me limits of 1 and 0, so it's not something I can use)



Maybe you guys can give me some ideas or theorems that can be of use here? I know that this problem could be probably solved using some more advanced calculus techniques, but I'm not familiar with derivatives and integrals yet, and this problem is in the limits section of my problem book, so I'm not supposed to use anything more advanced.


Answer



For every 1kn, n2n2+kn2+n hence the nth sum Sn is such that

nk=1kn2+nSnnk=1kn2.
Now, if it happens that the lower bound and the upper bound both converge to the same limit , then limnSn=.



Hence, what is left to do is to show that exists and to compute its value (or rather, to check that your guess that =12 is indeed correct--which it is).


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