Determine whether the sequence converges or diverges. If it converges, find the DNE.)
$$
\begin{array}{l}
a_{n}=\frac{n}{5} \sin (5 / n) \
\lim _{n \rightarrow \infty} a_{n}=\square
\end{array}
$$

Question

Determine whether the sequence converges or diverges. If it converges, find the DNE.)
$$
\begin{array}{l} 
a_{n}=\frac{n}{5} \sin (5 / n) \
\lim _{n \rightarrow \infty} a_{n}=\square
\end{array}
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start by analyzing the given sequence $a_n = \frac{n}{5} \sin \left( \frac{5}{n} ight)$. ‖ ‖ step 2 ⋮ To find the limit as $n$ approaches infinity, we use the fact that $\sin(x) \approx x$ when $x$ is close to 0. Here, as $n o \infty$, $\frac{5}{n} o 0$. ‖ ‖ step 3 ⋮ Substitute $\sin \left( \frac{5}{n} ight) \approx \frac{5}{n}$ into the sequence: $a_n \approx \frac{n}{5} \cdot \frac{5}{n}$. ‖ ‖ step 4 ⋮ Simplify the expression: $a_n \approx \frac{n}{5} \cdot \frac{5}{n} = 1$. ‖ ‖ step 5 ⋮ Therefore, $\lim_{n o \infty} a_n = 1$. ‖ ∻Answer∻ ⚹ The sequence converges to 1. ⚹ ∻Key Concept∻ ⚹Limit of a sequence involving trigonometric functions⚹ ∻Explanation∻ ⚹By approximating $\sin(x)$ with $x$ for small values of $x$, we can simplify the sequence and find its limit as $n$ approaches infinity.⚹◊What is the limit of $a_n$ as $n$ approaches infinity for the sequence $a_n = \frac{n}{5} \sin\left(\frac{5}{n} ight)$?⍭ Generate me a similar question◊

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