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Part CCSIR NET December 2024whether-the-factor-vanishes-at-the-endpoint

Whether the factor vanishes at the endpoint

Consider the following functions on the interval [0,1]:g(x)=sin⁡(πx),h(x)=cos⁡(π(x−1))[0, 1] : g(x) = \sin(\pi{}x), h(x) = \cos(\pi(x - 1)). Which of the following statements are true?

  1. A.(xng(x))n≥1(x^{n}g(x))_{n}\ge_{1} converges uniformly on [0, 1].✓ Correct
  2. B.(xng(x))n≥1(x^{n}g(x))_{n}\ge_{1} does not converge uniformly on [0, 1].
  3. C.(xnh(x))n≥1(x^{n}h(x))_{n}\ge_{1} converges uniformly on [0, 1].
  4. D.(xnh(x))n≥1(x^{n}h(x))_{n}\ge_{1} does not converge uniformly on [0, 1].✓ Correct

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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: fₙ → f uniformly ⇒ fₙ′ → f′

More on this topic

The chapter behind this: Uniform convergence — what it buys and how to test it — free to read

From Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini

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