Subtiers or Series Limit Type Tier Expression Notes Examples Convergence Two examples Super Large an = n" N/A among many An = n! т" 3n Exponentially Large Am = q", lq| > 1 Larger |q|, larger tier en (-2)" (3/2)" ne lim an = ∞ n2 n=1 An = n°, s > 0 Larger s, larger tier Positive Power Diverges Vn n?/3 [In(n)]? Positive An = [In(n)]ª, Larger s, larger tier Logarithmic In(n) Power [In(n)]1/2 and 1/an 2n e.g. an = C + 0, an = (-1)", an = Both An Bounded are bounded n +1 Negative Logarithmic an = [In(n)]®, s < 0 Larger s, larger tier [In(n)]¬1/2 1/ In(n) ±an n=1 [In(n)]-2 1//n Power Diverges Or Converges Conditionally* Negative Power Larger s, larger tier An = n°, Twilight Realm* 1.0000001 -2 п lim an = 0 (1/2)" e-n Exponentially Small Larger |q|, larger tier an = q", An 0 < [q| < 1 1/(-3)" n=1 Converges Absolutely Super Small Two examples an = e" /n! N/A among many ат, — п Zero Smallest With the help of the "tierlist", sort the following sequences in descending order: dn (-1)" + 3/In(n) bn n' +n-" An Сп en fn In п e" — (-4)" | т" sin(-n) + (-1)" п-е е—п u-

Subtiers or Series Limit Type Tier Expression Notes Examples Convergence Two examples Super Large an = n" N/A among many An = n! т" 3n Exponentially Large Am = q", lq| > 1 Larger |q|, larger tier en (-2)" (3/2)" ne lim an = ∞ n2 n=1 An = n°, s > 0 Larger s, larger tier Positive Power Diverges Vn n?/3 [In(n)]? Positive An = [In(n)]ª, Larger s, larger tier Logarithmic In(n) Power [In(n)]1/2 and 1/an 2n e.g. an = C + 0, an = (-1)", an = Both An Bounded are bounded n +1 Negative Logarithmic an = [In(n)]®, s < 0 Larger s, larger tier [In(n)]¬1/2 1/ In(n) ±an n=1 [In(n)]-2 1//n Power Diverges Or Converges Conditionally* Negative Power Larger s, larger tier An = n°, Twilight Realm* 1.0000001 -2 п lim an = 0 (1/2)" e-n Exponentially Small Larger |q|, larger tier an = q", An 0 < [q| < 1 1/(-3)" n=1 Converges Absolutely Super Small Two examples an = e" /n! N/A among many ат, — п Zero Smallest With the help of the "tierlist", sort the following sequences in descending order: dn (-1)" + 3/In(n) bn n' +n-" An Сп en fn In п e" — (-4)" | т" sin(-n) + (-1)" п-е е—п u-

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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