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I've tried to prove that the sequence is convergent by using the monotonic sequence theorem but after computing the first few terms, I realized that the sequence is not monotonic thus, it isn't possible to use the monotonic sequence theorem.

Are there any other ways to prove that the sequence is convergent? I'd really appreciate it if anyone can give me a hint or two on how to solve the question. Thanks!

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2 Answers 2

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Perhaps the sequence $a_1,a_3,a_5,...$ is monotone, and also $a_2,a_4,a_6,...$

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$\textbf{HINT:}$ Try checking the secuence $\{a_{2k}\}_{k\in\mathbb{N}}$ and $\{a_{2k-1}\}_{k\in\mathbb{N}}$. One of them is crecent and the other decrecent. But it converges iff $\limsup=\liminf$

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