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 $[0, 1] : g (x) = sin (π x), h (x) = cos (π (x - 1))$ . Which of the following statements are true?

A. $(x^{n} g (x))_{n} \geq_{1}$ converges uniformly on [0, 1].✓
B. $(x^{n} g (x))_{n} \geq_{1}$ does not converge uniformly on [0, 1].
C. $(x^{n} h (x))_{n} \geq_{1}$ converges uniformly on [0, 1].
D. $(x^{n} h (x))_{n} \geq_{1}$ does not converge uniformly on [0, 1].✓

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.

50 are analysed free — try those first.

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

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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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