수열의 수렴과 발산

First, we simplify. Remember $n! = n(n-1)(n-2)\cdots(2)(1)$ for any whole number $n$, so $(k+1)!=(k+1)k!$ .

$$\begin{aligned} a_k&=\frac{k!\sin^3k}{(k+1)!}=\frac{k!\sin^3k}{(k+1)k!} = \frac{\sin^3 k}{k+1} \end{aligned}$$

Now, we can use the squeeze theorem.

• $-1 \leq \sin k \leq 1$ for all $k$, so $-1 \leq \sin^3k \leq 1$. Let $b_k = \frac{-1}{k+1}$ and $c_k = \frac{1}{k+1}$. Then $b_k \leq a_k \leq c_k$.
• Both $\lim\limits_{k \to \infty}b_k=0$ and $\lim\limits_{k \to \infty}c_k=0$.

So, by the squeeze theorem, also $\displaystyle\lim_{k\to\infty} a_k=0$.