let-give-u-n-k-1-n-sin-k-n-sin-k-n-2-1-prove-that-the-sequence-u-n-is-convergent-2-find-lim-n-u-n-

Tinku Tara June 4, 2023 Limits 0 Comments

Question Number 31461 by abdo imad last updated on 08/Mar/18

let give u_n = Σ_(k=1) ^n sin ((k/n)) sin((k/n^2 )) 1) prove that the sequence (u_n ) is convergent 2) find lim_(n→∞) u_n .

$${let}\:{give}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\:\left(\frac{{k}}{{n}}\right)\:{sin}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{the}\:{sequence}\:\left({u}_{{n}} \right)\:{is}\:{convergent} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} . \\ $$

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