Thursday, 24 August 2017

Calculate the limit of the sequence limnrightarrowinftyan,ngeqslant1



Calculate the limit of the sequence




lim



knowing that



\ a_n = \frac{3^n}{n!},n\geqslant1



Choose the right answer:



a) 1




b) 0



c) 3



d) \frac{1}{3}



e) 2



f) \frac{1}{2}


Answer




Using D'Alambert's criterion, we can see that



\lim_{n \to \infty}\frac{a_{n+1}}{a_n}=\lim_{n \to \infty} \frac{3^{n+1}n!}{3^{n}(n+1)!}= \lim_{n \to \infty} \frac{3}{n+1}=0



Thus, \lim\limits_{n \to \infty} a_n =0.


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