Question

If in a sequence {a_n}, a_n = n! / nⁿ, then the sequence converges to
1. zero
2. one
3. two
4. none of these

Answer 1

Correct Answer - Option 1 : zero

Calculation:

We have, 

⇒ \(a_n = \frac{1.2.3...n}{n.n.n...n}\)

⇒ \(a_n=(\frac{1}{n})(\frac{2}{n})(\frac{3}{n})...(\frac{n}{n})<\frac{1}{n}\)

Thus, 0 < a_n < 1/n

Taking limit as n-->∞, we have

⇒ 0 ≤ \(\mathop {\lim }\limits_{n \to \infty } a_n\) ≤ \(\mathop {\lim }\limits_{n \to \infty } (\frac{1}{n})\) = 0

Therefore, \(\mathop {\lim }\limits_{n \to \infty } a_n\) = 0

Hence the sequence  converges to zero.