Part BCSIR NET December 2024split-the-term-before-applying-the-m-test

Split the term before applying the m test

Consider the sequences $(f_{n})_{n} \geq_{1}$ and $(g_{n})_{n} \geq_{1}$ of functions defined on the interval [−1, 1] by $f_{n} (x) = (- 1)^{n} (x^{2} + n) / n^{2}$ and $g_{n} (x) = (- 1)^{n} (x^{2} + n^{2}) / n^{3}$ . Which of the following statements is true?

A. $\sum (n \geq 1) f_{n}$ and $\sum (n \geq 1) g_{n}$ are uniformly convergent on the interval [−1, 1].✓
B. $\sum (n \geq 1) f_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum (n \geq 1) g_{n}$ is not.
C. $\sum (n \geq 1) g_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum (n \geq 1) f_{n}$ is not.
D.Neither $\sum (n \geq 1) f_{n}$ nor $\sum (n \geq 1) g_{n}$ is uniformly convergent on the interval [−1, 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.

See pricing

50 are analysed free — try those first.

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

Split the term before applying the m test

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