wk12solns - MA104 Week 12 Report Ratio/Root Tests and...

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Unformatted text preview: MA104 Week 12 Report & Ratio/Root Tests and Absolute/Conditional Convergence; Power Series Name: Fall 2010 Student Number: Lab: 1. [6 marks ] In the early 1900s, mathematician Srinivasa Ramanujan discovered the series: s = p 8 9801 1 X n =0 (4 n )!(1103 + 26390 n ) 396 4 n ( n !) 4 ! & 1 (a) Use the ratio test to show that the series 1 X n =0 a n = 1 X n =0 (4 n )!(1103 + 26390 n ) 396 4 n ( n !) 4 converges absolutely. lim n !1 & & & & a n +1 a n & & & & = lim n !1 & & & & (4( n + 1))!(1103 + 26390( n + 1)) 396 4( n +1) (( n + 1)!) 4 & ¡ 396 4 n ( n !) 4 (4 n )!(1103 + 26390 n ) ¢& & & & = lim n !1 & & & & (4 n + 4)!(27493 + 26390 n ) 396 4 n +4 (( n + 1)!) 4 & ¡ 396 4 n ( n !) 4 (4 n )!(1103 + 26390 n ) ¢& & & & = lim n !1 & & & & (4 n + 4)(4 n + 3)(4 n + 2)(4 n + 1) 396 4 ( n + 1) 4 ¡ 27493 + 26390 n 1103 + 26390 n ¢& & & & = 1 396 4 lim n !1 & & & & ¡ 4 n + 4 n + 1 ¢¡ 4 n + 3 n + 1 ¢¡ 4 n + 2 n + 1 ¢¡ 4 n + 1 n + 1 ¢¡ 27493 + 26390 n 1103 + 26390 n ¢& & & & = 1 396 4 lim n !1 & & & & ¡ 4 + 4 n...
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