```
Determine whether the sequence converges or diverges. If it converges, $f$ DNE.)
$\begin{array}{l}
\{0,4,0,0,4,0,0,0,4, \ldots\} \\
\lim _{n \rightarrow \infty} a_{n}=\square
\end{array}$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

To determine whether the sequence $\{0, 4, 0, 0, 4, 0, 0, 0, 4, \ldots\}$ converges or diverges, we need to find the limit of $a_n$ as $n$ approaches infinity

step 2

Observe that the sequence does not settle into a single value as $n$ increases. Instead, it continues to oscillate between 0 and 4

step 3

Since the sequence does not approach a single value, it does not converge

step 4

Therefore, the limit $\lim_{n \rightarrow \infty} a_n$ does not exist (DNE)

Answer

The sequence diverges.

Key Concept

Divergence of a sequence

Explanation

A sequence diverges if it does not approach a single value as $n$ approaches infinity. In this case, the sequence $\{0, 4, 0, 0, 4, \ldots\}$ oscillates and does not settle into a single value, hence it diverges.

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