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I have just started reading the book "Measure, Integral and Probability" 2nd ed. by Marek Capinski and Ekkehard Kopp.

The book starts out with a discussion on the Riemann Integral, its scope and limitations. An example is given.

The example results in an upper Riemann sum as:

$\displaystyle \frac{1}{n^3}\sum_{i=1}^n ({2i^2 - i})$

The authors then state that the this sum converges to a value and that this is easily seen.

How can i find the value to which this sum converges?

HenrikJson
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  • There are well known formulas for sums of powers - you could use these and explicitly evaluate the sum in terms of $n$. – Guy Paterson-Jones Oct 30 '16 at 19:56
  • Have you done a course on basic real - analysis or elementary algebra? If not, I would suggest you to start there, before studying this book. – IamThat Oct 30 '16 at 19:59
  • yes, i have done those courses. might be something got lost on the way though as it was long time ago. Any more concrete advice would be highly appreciated – HenrikJson Oct 30 '16 at 20:53
  • the power sum would have the summing variable as exponent. in the sum provided in the book the summing variable is in the base – HenrikJson Oct 30 '16 at 20:56

1 Answers1

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I have found a bunch of other posts that deals with related problems.

I still do not want to close mine as duplicate, just highlight the following posts, which can be applied with minor adaptions to solve the problem:

Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially

Prove that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?

Community
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HenrikJson
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