Chapter 9.1, Problem 53E

To determine

To Calculate: The limit of a sequence.

Answer to Problem 53E

The sequence converges to the limit $3\pi$ .

Explanation of Solution

Given Information: The limit to be calculated is $\underset{n\to \infty }{\lim }\left[n\sin \left(\frac{3\pi }{n}\right)\right]$

Concept Used:

• $\underset{\theta \to 0}{\lim }\frac{\sin \theta }{\theta }=1$
• If $\underset{x\to c}{\lim }f\left(x\right)=M$ , then $\underset{x\to c}{\lim }\left[k\cdot f\left(x\right)\right]=k\cdot M$(Constant Multiple Rule)

Calculation: In $\underset{n\to \infty }{\lim }\left[n\sin \left(\frac{3\pi }{n}\right)\right]$ , replace $\frac{3\pi }{n}$ with $\theta$ to write

  $\underset{n\to \infty }{\lim }\left[n\sin \left(\frac{3\pi }{n}\right)\right]=\underset{\frac{3\pi }{n}\to 0}{\lim }\left[\frac{3\pi \cdot \sin \left(\frac{3\pi }{n}\right)}{\left(\frac{3\pi }{n}\right)}\right]$

  $=\underset{\theta \to 0}{\lim }\left(\frac{3\pi \cdot \sin \theta }{\theta }\right)$

  $=3\pi \cdot \underset{\theta \to 0}{\lim }\left(\frac{\sin \theta }{\theta }\right)$ (Constant Multiple Rule)

  $=3\pi \cdot 1=3\pi$

Therefore, the sequence converges to the limit $3\pi$ .