The problem asks us to determine if the series $\sum_{k=1}^{\infty} k \sin \frac{1}{k}$ converges or diverges.

1. Problem Description

The problem asks us to determine if the series $\sum_{k=1}^{\infty} k \sin \frac{1}{k}$ converges or diverges.

2. Solution Steps

We will use the limit test for divergence. If $\lim_{k \to \infty} a_k \neq 0$, then the series $\sum a_k$ diverges.

In our case, $a_k = k \sin \frac{1}{k}$.

We want to find $\lim_{k \to \infty} k \sin \frac{1}{k}$.

Let $x = \frac{1}{k}$. As $k \to \infty$, $x \to 0$. Thus,

$$\lim_{k \to \infty} k \sin \frac{1}{k} = \lim_{x \to 0} \frac{\sin x}{x}$$

We know that $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

Thus,

$$\lim_{k \to \infty} k \sin \frac{1}{k} = 1$$

Since $\lim_{k \to \infty} k \sin \frac{1}{k} = 1 \neq 0$, the series $\sum_{k=1}^{\infty} k \sin \frac{1}{k}$ diverges by the divergence test.

3. Final Answer

Diverges