Express sum as a definite integral

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SUMMARY

The limit expression \(\lim_{n\to\infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^2+\left(\frac{2}{n}\right)^2+\cdots+\left(\frac{n-1}{n}\right)^2\right]\) can be expressed as the definite integral of the function \(f(x) = x^2\) over the interval [0, 1]. This conclusion is derived from the understanding of Riemann sums, which approximate the area under a curve and converge to the integral as \(n\) approaches infinity. The integral representation is \(\int_0^1 x^2 \, dx\), which evaluates to \(\frac{1}{3}\).

PREREQUISITES
  • Understanding of Riemann sums
  • Basic knowledge of definite integrals
  • Familiarity with limits in calculus
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study Riemann sums and their relationship to definite integrals
  • Learn how to compute definite integrals using fundamental theorems of calculus
  • Explore the properties of continuous functions and their integrals
  • Practice problems involving limits and their applications in calculus
USEFUL FOR

Students studying calculus, particularly those learning about integrals and limits, as well as educators seeking to explain the connection between Riemann sums and definite integrals.

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Homework Statement

If n is a positive integer, then
[tex]\lim_{n\to\infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^2+\left(\frac{2}{n}\right)^2+\cdot\cdot\cdot+\left(\frac{n-1}{n}\right)^2\right][/tex]can be expressed by what definite integral?

The attempt at a solution

A student I was helping had this problem and I had no idea how to even start. It was a problem along with other basic calc I definite and indefinite integrals, so I'm guessing it has some easy solution that I'm completely missing.
 
Physics news on Phys.org
You don't know Riemann sums as approximations to integrals??
 
I knew it was something simple :rolleyes:
Makes sense; now I just need to figure out f(x)...
 

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