Evaluate $$\lim$_{n$\to\]\infty$}a_{n}$ and determine whether the sequence converges or diverges.

$a_{n}=5$\cos\[\left$($\frac{14}{n^2}\]\right$)$

$a_{n}=5$ , converges.

$a_{n}=5$ , diverges.

$a_{n}=DNE$ , converges.

$a_{n}=DNE$ , diverges.

Verified step by step guidance

Identify the sequence given: a_n = 5$\cos\[\left$($\frac{14}{n^2}\]\right$).

Understand that as n approaches infinity, the term $\frac{14}{n^2}$ approaches 0.

Recall that $\cos$(0) = 1, so as $\frac{14}{n^2}$ approaches 0, $\cos\[\left$($\frac{14}{n^2}\]\right$) approaches $\cos$(0) = 1.

Substitute the limit into the sequence: a_n = 5 $\times$ 1 = 5.

Conclude that the sequence converges to 5 as n approaches infinity.

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Evaluate limn→∞an\(\lim\)_{n\(\to\]\infty\)}a_{n}limn→∞an and determine whether the sequence converges or diverges.an=5cos(14n2)a_{n}=5\(\cos\[\left\)(\(\frac{14}{n^2}\]\right\))an=5cos(n214)