Exercise · ZW-CAUCHY-SEQUENCE-006 · ★★☆ Intermediate
Let \(H_n=1+\frac12+\cdots+\frac1n\). What is the smallest value of \(|H_{2n}-H_n|\) over \(n\ge1\)?
Choose one option.
Show answer
- D. \(\tfrac12\)
Explanation
\(H_{2n}-H_n\) is a sum of \(n\) terms, each at least \(\frac1{2n}\), and it increases with \(n\); its minimum is at \(n=1\): \(\frac12\). So \(H_n\) is not Cauchy even though \(\frac1n\to0\).
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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