Mathematics Post Does the following sequence an=cosn/n^2 converge, or does it diverge? Find the limit if it is a convergent sequence. ByMeruprastaar 2022-04-282022-05-02Write a Comment on Does the following sequence an=cosn/n^2 converge, or does it diverge? Find the limit if it is a convergent sequence. Share Table of Contents Toggle 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^2a_n=\frac{cos\ n}{n^2} Recall that −1 ≤ cos(n) ≤ 1Hence\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}=0we conclude, by the Sandwich Principle, that the sequence {a_n} converges to the same limit:\lim_{n\rightarrow \infty}a_nKeywordsconvergence of sequence examplesconvergence of sequence and series 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 Share Related If Σ an and Σ bn are series of positive terms and lim an/bn=0 from n to +∞ show by example that Σ an may converge and Σ bn not converge solved problems in calculus
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