Part CCSIR NET June 2024splitting-a-conditionally-convergent-series

Splitting a conditionally convergent series

Let $\sum_{n}$ ₌ $_{1}^\infty a_{n}$ be a convergent series of real numbers. For n ≥ 1 define $A_{n} = a_{n}$ if $a_{n} > 0$ and 0 otherwise; $B_{n} = a_{n}$ if $a_{n} < 0$ and 0 otherwise. Which of the following statements are necessarily true?

A. $A_{n} \to 0$ and $B_{n} \to 0$ as $n \to \infty$ ✓ Correct
B.If $\sum_{n}$ ₌ $_{1}^\infty a_{n}$ is absolutely convergent, then both $\sum_{n}$ ₌ $_{1}^\infty A_{n}$ and $\sum_{n}$ ₌ $_{1}^\infty B_{n}$ are absolutely convergent✓ Correct
C.Both $\sum_{n}$ ₌ $_{1}^\infty A_{n}$ and $\sum_{n}$ ₌ $_{1}^\infty B_{n}$ are convergent
D.If $\sum_{n}$ ₌ $_{1}^\infty a_{n}$ is not absolutely convergent, then both $\sum_{n}$ ₌ $_{1}^\infty A_{n}$ and $\sum_{n}$ ₌ $_{1}^\infty B_{n}$ are divergent✓ 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 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: Σaₙ convergent ⇒ Σaₙ² convergent

Splitting a conditionally convergent series

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