L22 Absolute Convergence Ratio and Root Tests Part II - verges 7 1 X n =0 1 n arctan n n 2(conv absolutely by CT 1 1 3 3 1 3 5 5 1 n 1 1 3 5(2 n 1(2 n

# L22 Absolute Convergence Ratio and Root Tests Part II -...

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Lecture 22: Absolute Convergence and Ratio and Root Tests (II) The Root Test- Let X j a n j be a series. 1. If lim n !1 n p j a n j = L < 1 ; then the series is absolutely convergent. 2. If lim n !1 n p j a n j = L > 1 ; or lim n !1 n p j a n j = 1 , then the series is divergent. 3. If lim n !1 n p j a n j = 1 ; then the Root Test is inconclusive. 1
ex. 1 X n =1 n 2 n + 5 n ex. 1 X n =1 1 + 1 n n 2 2
Remark: lim n !1 n p n = 1 ex. 1 X n =2 n (ln n ) n 3
ex. 1 X n =1 cos( 1 n ) n 1 n ex. 1 X n =1 e 2 n n n 4
ex. 1 X n =1 (2 n )! ( n !) 2 5
ex. 1 X n =1 1 3 5 (2 n 1) (2 n )! 6
NYTI: 1 X n =1 sin( 1 n ) n (conv: LCT with b n = 1 n 2 ) 1 X n =1 (cos x ) 2 n 2 n (conv: geometric r = cos 2 x 2 ) 1 X n =3 ( 1) n ln n (conditional convergent) 1 X n =1 n ! n n (conv: Ratio , 1 e < 1 ) 1 X n =1 n + 1 n n 2 (div,root) Absolute or Conditional convergent? 1 X n =1 cos( n = 3) n ! (Conv: P 1 n ! con,Ratio ; j a n j = j cos( n = 3) j n ! 1 n ! , hence the original series converges Absolutelyby Comparison test, that implies the original series con- verges.) 7
1 X n =0 ( 1) n arctan n n 2 (conv absolutely by CT) 1 1 3 3! + 1 3 5 5! +( 1) n 1 1 3 5 (2 n 1) (2 n 1)! + (conv absolutely (hence conv.) by Ratio Test, lim j a n +1 a n j = 0) 1 X n =1 ( 1) n 2 n n ! 5 8 11 (3 n + 2) (conv absolutely by Ratio test, lim j a n +1 a n j = 2 3 < 1 ) 1 X n =1 3 n 2 + 5 2 n 2 + 2 n (div{Root) 1 X n =1 2 n 2 + 5 3 n 2 + 2 n (con-root) Tonight’s homework: Work out all NYTI problems. 8