Exercise · ZW-CONVERGENT-SEQUENCE-002 · ★☆☆ Basic
For \(a_n=\frac1n\) and \(\varepsilon=0.01\), what is the smallest \(N\) such that \(|a_n-0|<\varepsilon\) for all \(n\ge N\)?
Choose one option.
Show answer
- B. \(101\)
Explanation
We need \(1/n<0.01\), that is \(n>100\). The first integer that works is \(101\); for \(n=100\) we get equality, not strict inequality.
Review the concept: Convergent sequence
A sequence \((a_n)\) converges to a number \(L\) if, for every tolerance \(\varepsilon>0\), there is an index \(N\) from which all terms are within distance \(\varepsilon\) of \(L\).
\[\forall\varepsilon>0\ \exists N:\ n\ge N\Rightarrow|a_n-L|<\varepsilon\]
More practice problems: convergent sequence
- ZW-CONVERGENT-SEQUENCE-001 Compute lim _n→∞1n.
- ZW-CONVERGENT-SEQUENCE-003 Compute lim _n→∞(3n+1)/(n+2).
- ZW-CONVERGENT-SEQUENCE-004 What happens with the sequence a_n=(-1)^n?
- ZW-CONVERGENT-SEQUENCE-005 Compute lim _n→∞(2n^2+1)/(n^2+n).
- ZW-CONVERGENT-SEQUENCE-006 For a_n=(n+1)/(n), which converges to 1, and ε=0.1, what is the smallest N that works?
- ZW-CONVERGENT-SEQUENCE-007 If a_n→ L, how many terms of the sequence fall outside the interval (L-ε,L+ε), for a fixed ε>0?
- ZW-CONVERGENT-SEQUENCE-008 Compute lim _n→∞(√n+1-√ n).
- ZW-CONVERGENT-SEQUENCE-009 If a_n→ L, which statement is always true?