Exercise · ZW-CONVERGENT-SEQUENCE-009 · ★★★ Advanced
If \(a_n\to L\), which statement is always true?
Choose one option.
Show answer
- C. \(|a_n|\to|L|\)
Explanation
Since \(\big||a_n|-|L|\big|\le|a_n-L|\), \(|a_n|\to|L|\) holds. The others fail: \(1/a_n\) need not exist if \(a_n=0\) or \(L=0\); \(a_n=1/n\) has \(L=0\); and \(a_n=5-1/n\) has \(L=5\).
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-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).