$Probe the following sequences is null$

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$Probe the following sequences is null$

Use the definition of a null sequence to prove that the following sequence An(is null) An=(-1)^(n+1)/(8n^3 - 3) , n=1, 2...

N. b
An (the n is meant to be a small n)

Sequences and Series

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Let $\epsilon>0$. We want to file $N_0$ so that for all $n\geq N_0$ we have $|A_n|<\epsilon$, i.e.
\[|\frac{(-1)^{n+1}}{8n^3-3}|= \frac{1}{8n^3-3}<\epsilon\]
\[\Rightarrow 8n^3-3>\frac{1}{\epsilon} \Rightarrow n^3> \frac{1}{8}(\frac{1}{\epsilon}+3)\]
\[\Rightarrow n> \sqrt[3]{\frac{1}{8}(\frac{1}{\epsilon}+3)}. \]
So we can choose $N_0$ to be an integer such that
\[N_0> \sqrt[3]{\frac{1}{8}(\frac{1}{\epsilon}+3)},\]
then for $n>N_0$ we have
\[|A_n|<\epsilon.\]
This means that $A_n$ is a null sequence.

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$Probe the following sequences is null$

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