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Part BCSIR NET December 2023periodic-subsequence

Periodic subsequence

Consider the sequence $(a_{n})_{n \geq 1}$ , where $a_{n} = cos ((- 1)^{n} nπ /2 + nπ /3)$ . Which one of the following statements is true?

A. $lim sup a_{n} = 3 /2$ .
B. $lim sup a_{2 n} = 1$ .
C. $lim sup a_{2 n} = 1/2$ .
D. $lim sup a_{3 n} = 0$ .

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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: limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ

More on this topic

limits of perturbed argumentsJune 2023

From The Real Line › limsup, liminf and subsequential limits

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