Determine the limit of the sequence $a _ { n } = \frac { 5 \cos n + n } { n ^ { 2 } }$ .
A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)Divergent

Question

Quizplus · Accepted Answer

The limit of the sequence $a_n = \frac{5\cos n + n}{n^2}$ as $n$ approaches infinity can be found by analyzing the behavior of the numerator and the denominator separately. The term $5\cos n$ oscillates between -5 and 5, which becomes negligible compared to $n$ as $n$ grows. Thus, the dominant term in the numerator is $n$, and the denominator is $n^2$. The limit can then be simplified to $\lim_{n \to \infty} \frac{n}{n^2} = \lim_{n \to \infty} \frac{1}{n} = 0$.

Determine the Limit of the Sequence an=5cos⁡n+nn2a _ { n } = \frac { 5 \cos n + n } { n ^ { 2 } }an​=n25cosn+n​

Determine the Limit of the Sequence $a _ { n } = \frac { 5 \cos n + n } { n ^ { 2 } }$