Does the following sequence an=cosn/n^2 converge, or does it diverge? Find the limit if it is a convergent sequence.

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Does the following sequence a_n=\frac{cos\ n}{n^2}  converge, or does it diverge? Find the limit if it is a convergent sequence. Explain | an=cosn/n^2

a_n=\frac{cos\ n}{n^2}
Recall that −1 ≤ cos(n) ≤ 1

Hence

\frac{-1}{n^2}\leq \frac{cos\ n}{n^2}\leq\frac{1}{n^2}

Now since  \lim_{n\rightarrow \infty}\frac{-1}{n^2}=\lim_{n\rightarrow \infty}\frac{1}{n^2}=0

we conclude, by the Sandwich Principle, that the sequence {a_n} converges to the same limit:

\lim_{n\rightarrow \infty}a_n

Keywords

convergence of sequence examples

convergence of sequence and series


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