Exercise · ZW-MONOTONE-SEQUENCE-008 · ★★☆ Intermediate
The sequence \(a_n=\frac{2n}{n^2+1}\) is:
Choose one option.
Show answer
- D. strictly decreasing
Explanation
Its first terms are \(1,\ \tfrac45,\ \tfrac35,\ \tfrac8{17},\dots\) and \(a_n-a_{n+1}=\frac{2(n^2+n-1)}{(n^2+1)((n+1)^2+1)}>0\) for all \(n\ge1\).
Review the concept: Monotone sequence
A real sequence \((a_n)\) is monotone when, as the index \(n\) grows, its terms either never decrease or never increase.
\[a_n\le a_{n+1}\ \ \forall n\quad\text{or}\quad a_n\ge a_{n+1}\ \ \forall n\]
More practice problems: monotone sequence
- ZW-MONOTONE-SEQUENCE-001 The sequence a_n=2n+5 is:
- ZW-MONOTONE-SEQUENCE-002 The sequence a_n=1n is:
- ZW-MONOTONE-SEQUENCE-003 The sequence a_n=(-1)^n is:
- ZW-MONOTONE-SEQUENCE-004 The sequence a_n=(n)/(n+1) is:
- ZW-MONOTONE-SEQUENCE-005 Let a_n=n^2-6n. Which statement is correct?
- ZW-MONOTONE-SEQUENCE-006 Let a_1=-4 and a_n+1=(a_n)/(2). The sequence is:
- ZW-MONOTONE-SEQUENCE-007 If (a_n) is increasing and a_1=3, what always holds?
- ZW-MONOTONE-SEQUENCE-009 If (a_n) and (b_n) are increasing, which sequence is always increasing?