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Part BCSIR NET December 2024the-rate-of-a-riemann-sum-not-just-its-limit

The rate of a riemann sum not just its limit

Let f:[0,1]→Rf : [0, 1] \to \mathbb{R} be defined by f(x)=sin⁡(x2)f(x) = \sin(x^{2}). Let A=lim⁡(n→∞)(∑(k=1)nf(k/n)−n∫01f(x)A = \lim(n\to\infty) (\sum_(k=1)^n f(k/n) - n\int_{0}^{1}f(x)dx). Which of the following statements is true?

  1. A.A = 0
  2. B.A = 1
  3. C.A = sin(1)/2✓ Correct
  4. D.A = sin(1/4)

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Drill statements like this

Related counterexample: |f| Riemann integrable ⇒ f Riemann integrable

More on this topic

The chapter behind this: Riemann integrability — the criterion and the zoo — free to read

From Integration › Riemann integration and criteria

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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