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Real Analysis - Riemann Integrability

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Please see the attached file for the fully formatted problems.

Prove that if f is integrable on [0, 1], then
lim n !1 Z
1
0
x n f(x)dx = 0

Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition, upper and lower Riemann sums, (Riemann) integrability, upper and lower integrals in terms of the infimum and supremum of the respective sums, Riemann sums as the limit, properties of the integral, and the Mean Value Theorems for Integrals.

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Solution Summary

The solution is comprised of an explanation for using the Riemann integrability to determine the nature of a function. The solution is detailed and well presented.

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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