Question

Determine whether the sequence converges or diverges. If it converges, find the limit.

\({a_n} = \frac{{{{( - 3)}^n}}}{{n!}}\)

Short answer

Converges &.\(\mathop {\lim }\limits_{n \to \infty } {a_n} = 0\)

Step-by-step solution

Definition

A sequence\(\left\{ {{a_n}} \right\}\)has the limit \(L\)and we write \(\mathop {\lim }\limits_{n \to \infty } {a_n} = L\;\;or\;\;{a_n} \to L\)as\(n \to \infty \)if we can make the terms\({a_n}\)as close to\(L\)as we like by taking\(n\)sufficiently large. If \(\mathop {\lim }\limits_{n \to \infty } {a_n}\)exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

Evaluate limit

Consider the sequence\({a_n} = \frac{{{{( - 3)}^n}}}{{n!}}\).

It can be written as\(\frac{{{{( - 3)}^n}}}{{n!}} = \frac{{{{( - 1)}^n}{{(3)}^n}}}{{n!}}\)

We know that\(n! > {a^n}\)so

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

Thus given sequence is convergent and converges to 0.