Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.
A) $\frac { 1 } { 6 }$
B) $- 5$
C) $- \frac { 5 } { 6 }$
D) Diverges

Question

Quizplus · Accepted Answer

We can split the fraction in the following way:
$$a_n = \frac{n^2-5}{6n^2+1} = \frac{1}{6}-\frac{1}{6(6n^2+1)}$$
The second term approaches $0$ as $n$ grows larger since $6n^2+1$ increases much faster than $n^2$.

Thus, $\lim_{n	o\infty}a_n = \frac{1}{6}$, which means the sequence converges to $\frac{1}{6}$. Therefore, the best choice is A).

Determine Whether the Sequence Defined By an=n2−56n2+1a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }an​=6n2+1n2−5​

Determine Whether the Sequence Defined By $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$