Question

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

\({a_n} = \left\{ {\frac{{(2n - 1)!}}{{(2n + 1)!}}} \right\}\)

Short answer

Converges

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

Given a sequence\({a_n} = \left\{ {\frac{{(2n - 1)!}}{{(2n + 1)!}}} \right\}\).

Simplifying the sequence we have:

\(\begin{aligned}{a_n} &= \left\{ {\frac{{(2n - 1)!}}{{(2n + 1)!}}} \right\}\\ &= \frac{{(2n - 1)!}}{{2n(2n + 1)(2n - 1)!}}\\ &= \frac{1}{{2n(2n + 1)}}\end{aligned}\)

So, the limit is\(\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{2n(2n + 1)}}} \right) = 0\)

Thus the sequence converges to zero as \(n \to \infty \).