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yuanzhi's Question
Math
Posted 4 months ago

Determine whether the sequence converges or diverges. If it converges, ff DNE.)
{0,4,0,0,4,0,0,0,4,}limnan=
\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,}\{0, 4, 0, 0, 4, 0, 0, 0, 4, \ldots\} converges or diverges, we need to find the limit of ana_n as nn approaches infinity
step 2
Observe that the sequence does not settle into a single value as nn 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 limnan\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 nn approaches infinity. In this case, the sequence {0,4,0,0,4,}\{0, 4, 0, 0, 4, \ldots\} oscillates and does not settle into a single value, hence it diverges.

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