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(Requires calculus) Show that if c > b > 1. then bn is

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 60E Chapter 3.2

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 60E

(Requires calculus) Show that if , then  is but  is not .

The following problems deal with another type of asymptotic notation, called little-o notation. Because little-o notation is based on the concept of limits, a knowledge of calculus is needed for these problems.We say that is [read  is “little-oh” of ], when

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Week 7 MUL2010 Intro to Music Lit ▯1 Introduction to Music Literature ♪ WEEK 7 NOTES ♪ ▯ Main Theme - BEETHOVEN Ludwig van Beethoven (1770-1827) ⁃ Childhood ⁃ Born in Bonn, Germany ⁃ Father was persistent on his talents, made him practice at early hours ⁃ Mozart: “Keep your eyes on him; someday he will give the world something to talk about) ⁃ Mother died of TB, father was an alcoholic, took care of two brothers. ⁃ Age 22; moved to Vienna to study with Haydn ⁃ Why couldn’t Beethoven find his teacher Because he was Haydn ⁃ Counterpoint and fugue was outdated, but his main interest ⁃ Believed artists to be equal with nobles - rise in social status ⁃ Deafness ⁃ started at age 29 ⁃ reclusive; moved to Heiligenstadt ⁃ contemplated suicide ⁃ Fully independent musician ⁃ Age 44; declining health & personal turmoil ⁃ forced to stop performing, kept conducting ⁃ brother died in 1815, younger brother Carl under his care ⁃ 1809, threatened to leave vienna but was given an annual salary to not leave vienna ⁃ 1818, golden era, although he is completely deaf • His music - perfectionist, longer compositional process ⁃ Brought emotional power to classical music; transitional composer to romantic era ⁃ momentum from rhythmic ideas ⁃ expansion of tessitura ⁃ Unity in multi movement forms; ⁃ connects movements without pause ⁃ thematic material throughout movements ⁃ developmental sections longer ⁃ elongated codaic passages affirm victory of tonic key over the new keys stated in the development ⁃ climactic and triumphant finales ⁃ FORMS chappythewizard.tumblr.com Week 7 MUL2010 Intro to Music Lit ▯2 ⁃ SYMPHONIES - most popular works, written for larger orchestras than most in classical era with complex part writing. Odd are calmer, Even are more assertive, aggressive/forceful. ⁃ III - Eroica ⁃ Originally named after Napoleon and his ability to enable the people of France, but upon learning of his self appointed title of dictator, Beethoven renamed it. ⁃ V - Symphony no. 5 in C minor, Op. 67 (1808) ⁃ most popular: solsolsol MEEEE fafafa REEEE “fate knocking at the door” ⁃ I. allegro con brio - exploitation of motive in both themes ⁃ II. andante con moto - in major key; calm, lyrical ⁃ III. scherzo - opening motive returns, B section is a rapid, fugal theme ⁃ no break between I and II ⁃ IV. allegro - sonata form w/ extended coda and climactic expansion of the voices in the orchestra ⁃ VI - Pastoral ⁃ IX - An die Freude ⁃ includes chorus and vocal soloists ⁃ 32 Piano Sonatas display piano virtuosity - massive chords, passages where hands are widely separated ⁃ Piano Sonata in C minor, Op. 13 (“Pathétique”) ⁃ relative key of Eb major included ⁃ I. Grave; allegro molt e con brio - contrasting directions of feeling, generally v melancholy piece ⁃ six note motive; unresolved questions (dooooo, dore me meeeeee re, faaaaaa, fasol le leeeee sol) ⁃ constantly switching between a happy and sad feeling; melancholy, but sensitive ⁃ staccato motive in exposition ⁃ return to original theme in coda; simplified almost ⁃ Movement 1: Basically it’s the musical embodiment of a really bad cry sesh: you start off angry frustrated because you don’t know what you’re doing with your life and it’s like 10:30 pm on a sunday, then you get hysterical, then you get really sad, then you start throwing things because you don’t know how to feel, and you end up chappythewizard.tumblr.com Week 7 MUL2010 Intro to Music Lit ▯3 being emotionally drained and gloomy because not only do u feel bad but now your room’s a mess and you just want to take a nap and wallow in self pity. ⁃ II. Adagio Cantabile - Ab major - related to C minor ⁃ Rondo form: ABACA coda ⁃ A - legato, lyrical, middle section ⁃ B - legato, repeated chords ⁃ C - duet between hands in minor mode, feat. triplets ⁃ coda - descending chords, gentle ending ⁃ Movement 2: You took a sad nap and now you’re still sad but at least you know it’s gonna be okay later, but everything feels heavy because you just woke up and it’s a chilly fall afternoon. It’s rainy and you’re stretching around the house, maybe longing for something better. Lotsa sighing, maybe feelin a bit sad again with that B section coming in, but you feel supa sad with that C section, maybe because u were looking outside and theres a dog and he’s all alone in the rain, but then the owner comes to get him and it’s so cute that you happy cry (it’s not the same sad as before so you’re kinda watching a soap opera outside your window, mini roller coaster of emotions). You get up from the window because the dog is gone, and make urself some eggs and toast, and sit alone at a lil table eating ur ok breakfast and ur like “aw I kinda feel better now” bc those eggs taste like a HUG and a lil bit of hot cocoa. ⁃ III. Rondo (allegro) ⁃ Extended Rondo form: ABACABA coda ⁃ A section: Lively theme, c minor ⁃ B section: lyrical themes w dominant chord ⁃ C section: polyphonic, ideas shift between hands, ends on dominant chord ⁃ coda: idea from main theme ⁃ Movement 3: Reclamation the earlier sadness, working crazy hard but also kinda stressed but its chappythewizard.tumblr.com Week 7 MUL2010 Intro to Music Lit ▯4 ok bc you’re doing well and u just finished an essay or something so you get up and go get coffee or leave your general station to go do more stuff, but then come back to the same place, scene shifts to ur home w C and ur feelin pretty good all the time, chaos of you being you, basically just you going on with your life after that crazy huge cry sesh and mini quarter life crisis. LOLJK you THOUGHT coda comes in and as you trip UP (yes, up) the stairs spilling coffee all over yourself and your perf essay which you’ve been working on this entire movement, you remember life is suffering! fantastic ⁃ Piano Sonata in C# minor, Op. 27 no. 2 (Moonlight) ⁃ sonata quasi una fantasia ⁃ I. lyrical melody, arpeggios in accompaniment, strophic ish ⁃ rubato ⁃ String Quartets - highly sophisticated, break from traditional movement form ⁃ Piano Concertos - sometimes begin with pianos ⁃ One opera - fidelio ⁃ Compositional periods ⁃ early (to 1802) - more classical in nature, resembles Haydn and Mozart ⁃ middle (1803-1814) - increased power, heroism (after breakdown in Heiligenstadt) ⁃ late (1815 - 1827) - most dramatic, foreshadows romanticism ▯ ▯ Thanks for reading these study notes! Good luck! Any questions or comments visit chappythewizard.tumblr.com find more notes on Music Lit, Theory, and History on Studysoup! chappythewizard.tumblr.com

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Chapter 3.2, Problem 60E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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(Requires calculus) Show that if c > b > 1. then bn is