Subtiers or Series Limit Type Tier Expression Notes Examples Convergence Two examples Super Large an = n" N/A among many an = n! тп 3" Larger |q|, Exponentially Large An = q", |q| > 1 en larger tier (-2)" (3/2)" nº lim an = ∞ ±an n2 Positive n=1 Larger s, larger tier an = n°, п Power Diverges Vn n'/3 [In(n)]² In(n) Positive Logarithmic Power an = [In(n)]*, s > 0 Larger s, larger tier [In(n)]}/2 Both an and 1/an are bounded 2n Bounded e.g. an — C #0, ап — (-1)", аn %3D n+1 Negative Logarithmic Power In(n)]-1/2 1/ In(n) [In(n)]-2 1/Vn An = [In(n)]ª, Larger s, larger tier ±an n=1 Diverges Or Converges Conditionally" n-1 Negative Power Larger s, larger tier an =n°, Twilight Realm -1.0000001 n-2 lim an = 0 (1/2)" e-n Exponentially Small An = q", 0 < [q| < 1 Larger |q|, larger tier an 1/(-3)" n=1 Converges Absolutely Two examples an = e" /n! Super Small N/A among many An =n-n Zero An = 0 Smallest (2) With the help of the "tierlist", sort the following sequences in descending order: In dn fn en ba Сп e" - (-4)" | т'" | sin(-n) + (-1)" ап n° е—п n2 +n-" (-1)" + 3/In(n) п—е

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 35EQ
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Subtiers or
Series
Limit Type
Tier
Expression
Notes
Examples
Convergence
Two examples
Super
Large
an = n"
N/A
among many
an = n!
тп
3"
Larger |q|,
Exponentially
Large
An = q",
|q| > 1
en
larger tier
(-2)"
(3/2)"
nº
lim an = ∞
±an
n2
Positive
n=1
Larger s,
larger tier
an = n°,
п
Power
Diverges
Vn
n'/3
[In(n)]²
In(n)
Positive
Logarithmic
Power
an = [In(n)]*,
s > 0
Larger s,
larger tier
[In(n)]}/2
Both an and 1/an
are bounded
2n
Bounded
e.g. an — C #0, ап — (-1)", аn
%3D
n+1
Negative
Logarithmic
Power
In(n)]-1/2
1/ In(n)
[In(n)]-2
1/Vn
An = [In(n)]ª,
Larger s,
larger tier
±an
n=1
Diverges Or
Converges
Conditionally"
n-1
Negative
Power
Larger s,
larger tier
an =n°,
Twilight Realm
-1.0000001
n-2
lim an = 0
(1/2)"
e-n
Exponentially
Small
An = q",
0 < [q| < 1
Larger |q|,
larger tier
an
1/(-3)"
n=1
Converges
Absolutely
Two examples
an = e" /n!
Super
Small
N/A
among many
An =n-n
Zero
An = 0
Smallest
Transcribed Image Text:Subtiers or Series Limit Type Tier Expression Notes Examples Convergence Two examples Super Large an = n" N/A among many an = n! тп 3" Larger |q|, Exponentially Large An = q", |q| > 1 en larger tier (-2)" (3/2)" nº lim an = ∞ ±an n2 Positive n=1 Larger s, larger tier an = n°, п Power Diverges Vn n'/3 [In(n)]² In(n) Positive Logarithmic Power an = [In(n)]*, s > 0 Larger s, larger tier [In(n)]}/2 Both an and 1/an are bounded 2n Bounded e.g. an — C #0, ап — (-1)", аn %3D n+1 Negative Logarithmic Power In(n)]-1/2 1/ In(n) [In(n)]-2 1/Vn An = [In(n)]ª, Larger s, larger tier ±an n=1 Diverges Or Converges Conditionally" n-1 Negative Power Larger s, larger tier an =n°, Twilight Realm -1.0000001 n-2 lim an = 0 (1/2)" e-n Exponentially Small An = q", 0 < [q| < 1 Larger |q|, larger tier an 1/(-3)" n=1 Converges Absolutely Two examples an = e" /n! Super Small N/A among many An =n-n Zero An = 0 Smallest
(2) With the help of the "tierlist", sort the following sequences in descending order:
In
dn
fn
en
ba
Сп
e" - (-4)" | т'" | sin(-n) + (-1)"
ап
n°
е—п
n2 +n-"
(-1)" + 3/In(n)
п—е
Transcribed Image Text:(2) With the help of the "tierlist", sort the following sequences in descending order: In dn fn en ba Сп e" - (-4)" | т'" | sin(-n) + (-1)" ап n° е—п n2 +n-" (-1)" + 3/In(n) п—е
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