23. Intro to Derivatives & Area Under the Curve

Limits of Sequences

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Multiple Choice

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.

Master Limits of Sequences with a bite sized video explanation from Patrick

Evaluate lim⁡n→∞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​)