Exercise · ZW-CAUCHY-SEQUENCE-004 · ★☆☆ Basic
Let \(a_n=1+\dfrac{(-1)^n}{n}\). Compute \(|a_3-a_4|\).
Choose one option.
Show answer
- A. \(\tfrac7{12}\)
Explanation
\(a_3=1-\frac13=\frac23\) and \(a_4=1+\frac14=\frac54\), so the distance is \(\frac54-\frac23=\frac7{12}\).
Review the concept: Cauchy sequence
A sequence \((a_n)\) is Cauchy if, for every \(\varepsilon>0\), there is an index \(N\) beyond which any two terms are less than \(\varepsilon\) apart.
\[\forall\varepsilon>0\ \exists N:\ m,n\ge N\Rightarrow|a_m-a_n|<\varepsilon\]
More practice problems: cauchy sequence
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- ZW-CAUCHY-SEQUENCE-006 Let H_n=1+12+cdots+1n. What is the smallest value of |H_2n-H_n| over n≥1?
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- ZW-CAUCHY-SEQUENCE-008 Why is a_n=(-1)^n not Cauchy?
- ZW-CAUCHY-SEQUENCE-009 A Cauchy sequence (a_n) has a subsequence converging to L. What follows?