Chapter 11 - Infinite Sequences and Series - 11.1 Exercises: 24

1; Convergent

Work Step by Step

We have $\lim\limits_{n \to \infty}a_n = \lim\limits_{n \to \infty}\frac{n^3}{n^3+1}$. Dividing both the numerator and denominator by the highest term in the denominator, $n^3,$ we get that the limit equals $\lim\limits_{n \to \infty}\frac{n^3/n^3}{n^3/n^3+1/n^3}= \lim\limits_{n \to \infty}\frac{1}{1+1/n^3} =\frac{1}{1+0} =1.$ Therefore, $\lim\limits_{n \to \infty}a_n =1$ and the sequence $\{a_n\}$ converges.

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