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Part BCSIR NET December 2024a-discontinuous-sum-rules-out-uniform

A discontinuous sum rules out uniform

For integers n ≥ 0, let fn:[−1,0]→Rf_{n} : [-1, 0] \to \mathbb{R} be defined by fn(x)=x/(1−x)nf_{n}(x) = x/(1 - x)^{n}. Which of the following statements is true about the series ∑(n=0)∞fn\sum_(n=0)^\infty f_{n}?

  1. A.The series is neither absolutely convergent nor uniformly convergent.
  2. B.The series is both absolutely convergent and uniformly convergent.
  3. C.The series is absolutely convergent but not uniformly convergent.✓ Correct
  4. D.The series is uniformly convergent but not absolutely convergent.

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 pointwise vs uniform.

See pricing

50 are analysed free — try those first.

The trap it tests

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

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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