n=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n

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1 Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a a,. k= Defiitio If the sequece s of partial sums coverges to s R, we say that the series a coverges to s. We write a = s, ad we say that the umber s is the sum of the series a. If the sequece s diverges either lim s = ± or the limit does ot exist), we say that the series a diverges. Exercise Establish the followig idetities for x < : + x + x x = x+ x ; b) + x x = x+ + ) x + x ) ; c) x + x + x x = x+ + ) x + + x x ). Exercise Determie whether the followig series are coverget, ad if affirmative, determie the correspodig sums: b) c) e) f) g) h) i). Exercise Determie if the followig series are coverget, ad if affirmative determie the correspodig sums: b) + ) + ) + ) ; c) a R Z is a give umber) a + ) a + + ) e) + ) + ), f) + 4) g) 4 = ) + ) h) 4 i) l j) l + k) + 4). = Example The geometric series q coverges to q Exercise 4 Determie if the give sequece coverge/diverge: b) c) x, x < if q < ad diverges if q. why?) Exercise 5 Use the defiitio of the series to write. 7) = as a ratio of two itegers. Theorem Additio of series) If a ad b are coverget series, the a + b ) ad ca ) are also coverget series for ay c R), ad a + b ) = a + b,

2 respectively Exercise 6 FId the sum of the series ca ) = c a. +) + Theorem Cauchy s theorem for series) The series a coverges if ad oly if the sequece of partial sums is a Cauchy sequece, that is: for ay ε > 0 there exists N ε) such that ). s s m = a m a < ε, > m N ε). Propositio 4 If a coverges, the lim a = 0. Remark: the previous propositio is useful for showig that a give series is ot coverget: if lim a does ot exist or lim a 0, the the series a is diverget. Example The series ) ad + are diverget: for the first series lim ) does ot exist, ad for the secod series lim + = 0. Remark: the coverse of the previous propositio is ot true: if lim a = 0, the series a may or may ot be coverget! For example, cosider the series the series is diverget ad the secod oe is coverget. Alteratig series ad. For both series lim a = 0, but the first oe The followig theorem is useful whe dealig with series havig terms with alteratig ± sigs, called alteratig series: Theorem 5 Leibiz s Alteratig Series Theorem) If a a... 0 ad lim a = 0, the the series ) a coverges. Proof. Some algebra shows that s s m a m+ + a ), ) ad sice lim a = 0, we coclude that s ) is a Cauchy sequece. By Cauchy Theorem for series, we coclude that ) a coverges. satisfies the hypothe- Example Alteratig harmoic series) The alteratig harmoic series sis of Leibiz s theorem above, ad it is therefore coverget. ) Exercise 7 Use Leibiz s alteratig series theorem to show that the followig series are coverget.. ) + ) b) ) 6+0 Exercise 8 Cosider the sequece a = + ),. Show that a 0 for all, lim a = 0 but the series ) a diverges. Does this cotradict Leibiz s alteratig series theorem? Defiitio 6 We say that the series a is absolutely coverget if the series a is coverget. Remark: if a 0 for all, the absolute covergece is the same as covergece of the series, but i geeral they are differet.

3 Propositio 7 If a is absolutely coverget i.e. a is coverget), the a is coverget. ) Remark: the coverse of the propositio is ot true i geeral for example, the series is coverget but ot absolutely coverget). If a series is coverget but ot absolutely coverget, we say that it is coditioally coverget. Exercise 9 Determie if the give series are absolutely coverget/coditioally coverget/diverget: ) ) ) ) b) c) e) ) f) )!. Exercise 0 Determie if the series cos is coverget or diverget.. Series with o-egative terms: comparios test Cosider ow series with o-egative terms, that is series a for which a 0 for all. Propositio 8 If a 0 for, the series a coverges if ad oly the sequece s ) of partial sums is bouded. Example 4 Cosider the series!. Note that the sequece s ) of partial sums is bouded: s = < < + ) = + Sice a =! 0 for all, by the above theorem, the series! coverges. It ca be show that the sum of the series is equal to e Theorem 9 Compariso test with iequalities for series with o-egative terms) Let a be series with o-egative terms a, b 0,. i) If a b for all ad b coverges, the a coverges. ii) If a b for all ad b diverges, the a diverges. Remark: i short, for series with o-egative terms, smaller tha coverget is coverget, ad bigger tha diverget is diverget. Example 5 For example the series! coverges, because =! = = ad the series coverges geometric series with q = ). The series diverges, because ad the series diverges lim s = ). Exercise Determie whether the give series coverges or diverges. 5 l k b) c) k +. k= Exercise Use the compariso test to determie if the followig series are coverget. / + b) +5 c) cos π b

4 Theorem 0 Compariso test with limits for series with o-egative terms) Let a b be a series with positive terms a, b 0,. If lim b = c 0, ), the both series have the same ature both coverget or both diverget). Exercise Redo the previous two exercises usig the above compariso test with limits. Theorem Cauchy s Codesatio Theorem) If a a a... 0, the the series a coverges if ad oly if the series a = a + a + 4a 4 + 8a coverges. Corollary Covergece of the harmoic series) The series p. p coverges for p > ad diverges for. Three covergece tests Theorem D Alembert s Ratio Test) Give the series a a, suppose that l + a exists. i) If l <, the the series a coverges absolutely); ii) If l >, the the series a diverges; iii) If l =, the test is icoclusive the series may coverge or diverge). Example 6 Cosider the series. We have l a + a so by the Ratio test the series coverges absolutely) = <, Exercise 4 Determie if the give series coverge absolutely/coditially or diverge: ) b)!. Exercise 5 Use the Ratio test to determie if the followig series are coverget: b) )!)! 4)!! c) )!)! 4)!! )!)! 4)!!. Exercise 6 Show that ) ) coverges. Exercise 7 If a =... ) 4... ), show that the series a diverges, ad the series a coverges. Theorem 4 Cauchy s Root Test) Give the series a, suppose that l a exists more geerally, let l sup a ). i) If l <, the the series a coverges absolutely); ii) If l >, the the series a diverges; iii) If l =, the test is icoclusive the series may coverge or diverge). Example 7 Cosider the series. + ) We have so by the Root test the series l a + + ) coverges absolutely). ) + =, 4

5 Exercise 8 Test the covergece of the series. + +) Theorem 5 Raabe s test) Give the series ) a a, suppose that l a + exists. i) If l >, the the series a coverges absolutely); ii) If l <, the the series a diverges; iii) If l =, the test is icoclusive the series may coverge or diverge). Example 8 Cosider the series a, where a =... ) 4... ),. The Ratio test does ot apply, sice but usig Raabe s test we have lim a a + ad therefore the series a diverges. lim a + a + + =, ) ) = <, Additioal exercises. Determie whether the followig series are coverget π si b)!)/ c) ) + ) + ) e) )!) )! + / 4 +. Determie whether the followig series are coverget b) c) cos π) ) ) ), + + = + ; ; ),!!. f) )!!) ). Decide whether the followig series are absolutely coverget, coditioally coverget or diverget: b) c) = + + +)+) l = + l ) =! e) ; l + ) si ; f)) arcsi!! g)... ) ) h)... ) 4... ) i) al a > 0 is a parameter).... )!!!. 4. Show that if a coverges ad a 0 for all, the true without the hypothesis a 0? a 5. Show that if a coverges, the a p also coverges for ay p >. also coverges. Is the statemet 5