Identifying Sequence Convergence

A sequence $\{a_n\}$ is considered convergent if its terms approach a specific finite value as $n$ becomes infinitely large. This value is known as the limit of the sequence.

Calculating the Limit of $a_n$

To determine if the sequence defined by $a_n = \frac{n^2-2n}{3n^2+ n}$ converges, we calculate its limit as $n \to \infty$.

1. Divide the numerator and the denominator by the highest power of $n$ in the denominator, which is $n^2$: $a_n = \frac{\frac{n^2}{n^2}-\frac{2n}{n^2}}{\frac{3n^2}{n^2}+ \frac{n}{n^2}}$ $a_n = \frac{1-\frac{2}{n}}{3+\frac{1}{n}}$
2. Take the limit as $n \to \infty$. Since $\lim_{n \to \infty} \frac{k}{n} = 0$ for any constant $k$: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1-\frac{2}{n}}{3+\frac{1}{n}} = \frac{1-0}{3+0}$ $\lim_{n \to \infty} a_n = \frac{1}{3}$

Conclusion on Convergence

The limit of the sequence $a_n$ as $n$ approaches infinity is $\frac{1}{3}$. Since this limit is a finite value, the sequence $\{a_n\}$ is convergent.