# 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: Student Number: Spring 2012 1. [7 marks ] Consider the series s = 3 p 3 2 1 X n =0 ( n !) 2 (2 n + 1)! . (a) Use the Ratio Test to show that the series 1 X n =0 ( n !) 2 (2 n + 1)! converges absolutely. lim n !1 & & & & a n +1 a n & & & & = lim n !1 & & & & (( n + 1)!) 2 (2( n + 1) + 1)! & ¡ (2 n + 1)! ( n !) 2 ¢& & & & = lim n !1 & & & & (( n + 1)!) 2 (2 n + 3)! & ¡ (2 n + 1)! ( n !) 2 ¢& & & & = lim n !1 & & & & & ¡ ( n + 1)! n ! ¢ 2 & ¡ (2 n + 1)! (2 n + 3)! ¢ & & & & & = lim n !1 & & & & ( n + 1) 2 & ¡ 1 (2 n + 3)(2 n + 2) ¢& & & & = lim n !1 & & & & n + 1 2(2 n + 3) & & & & = lim n !1 & & & & 1 + 1 =n 4 + 6 =n & & & & = 1 4 < 1 ) By the ratio test, 1 X n =0 a n converges absolutely. (b) De&ne the partial sum s k = 3 p 3 2 k & 1 X n =0 ( n !) 2 (2 n + 1)! as a series in Maple: a:=(n)- > ???; s:=(k)- > (3*sqrt(3)/2)*sum(a(n),n=0..k-1); To 5 decimal places, evaluate (see Week 11 Lab Prep) each of the following using Maple.To 5 decimal places, evaluate (see Week 11 Lab Prep) each of the following using Maple....
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