Exercise · ZW-CONVERGENT-SEQUENCE-007 · ★★☆ Intermediate
If \(a_n\to L\), how many terms of the sequence fall outside the interval \((L-\varepsilon,L+\varepsilon)\), for a fixed \(\varepsilon>0\)?
Choose one option.
Show answer
- A. Finitely many
Explanation
From index \(N\) on all of them are inside; only \(a_1,\dots,a_{N-1}\) can be outside (sometimes none, but not always).
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-002 For a_n=1n and ε=0.01, what is the smallest N such that |a_n-0|<ε for all n≥ N?
- 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-008 Compute lim _n→∞(√n+1-√ n).
- ZW-CONVERGENT-SEQUENCE-009 If a_n→ L, which statement is always true?