Exercise · ZW-SUBSEQUENCE-010 · ★★★ Advanced
Let \(a_n=(-1)^n\left(1+\frac1n\right)\). What is the set of limits of its convergent subsequences?
Choose one option.
Show answer
- B. \(\{-1,1\}\)
Explanation
Even indices give \(1+\frac1{2k}\to1\) and odd ones \(-(1+\frac1{2k+1})\to-1\); any convergent subsequence has its limit in that set. The value \(0\) never appears.
Review the concept: Subsequence
A subsequence of a sequence \((a_n)\) is a sequence \((a_{n_k})\) made of terms of \((a_n)\) with strictly increasing natural indices \(n_k\).
\[a_{n_1},a_{n_2},a_{n_3},\dots\ \text{with}\ n_1
More practice problems: subsequence
- ZW-SUBSEQUENCE-001 Let a_n=n^2 and n_k=2k. What is the third term a_n_3 of the subsequence?
- ZW-SUBSEQUENCE-002 Which of these sequences is a subsequence of 1,tfrac12,tfrac13,tfrac14,dots?
- ZW-SUBSEQUENCE-003 Let a_n=(n+1)/(n). To what number does the subsequence a_k^2 converge?
- ZW-SUBSEQUENCE-004 Let a_n=(-1)^n. To what does the even-index subsequence a_2k converge?
- ZW-SUBSEQUENCE-005 Let a_n=3+1n. To what does the subsequence a_10^k converge?
- ZW-SUBSEQUENCE-006 Let a_n=sin ((nπ)/(2)). Which of these numbers is NOT the limit of any subsequence of (a_n)?
- ZW-SUBSEQUENCE-007 If a_2k→1 and a_2k+1→2, what happens to (a_n)?
- ZW-SUBSEQUENCE-008 Which of these index choices defines a subsequence (a_n_k) of any sequence (a_n)?