Evaluate `lim_(ntooo) sin^(n)((2pin)/(3n+1)),ninN.`

Explanation

To evaluate the limit \( \lim_{n \to \infty} \sin^n\left(\frac{2\pi n}{3n + 1}\right) \), we can follow these steps:

### Step 1: Simplify the Argument of the Sine Function
We start with the expression inside the sine function:
\[ \frac{2\pi n}{3n + 1} \]
As \( n \) approaches infinity, we can simplify this expression. We can factor out \( n \) from the denominator:
...