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Exercise 5 For all-natural number n, we define I, = F sin" x dx i.) a.) Calculate l, and /4. n+1 b.) Show that In+2 In n+2 c.) Show that (I,) is a positive decreasing sequence. d.) Using the preceding results, show that n+2 n+1 < !n+1 < 1. Hence, calculate the limit of 1. In ii.) We define the sequence t, = (n + 1)/n+1!n- a.) Show the sequence (t,) is a constant sequence. b.) Deduce that for all n, tn = iii.) We define v, = nl. a.) Express vn in terms of n, tn, In and In+1. Hence, calculate the limit of v. b.) Deduce the limit of nl. %3D iv.) a.) Calculate S sin 3x dx. 22 (nl)? (2n+1)! T(2n): b.) Show that In = and that I2n+1 = 22+ (n!)?

Exercise 5 For all-natural number n, we define I, = F sin" x dx i.) a.) Calculate l, and /4. n+1 b.) Show that In+2 In n+2 c.) Show that (I,) is a positive decreasing sequence. d.) Using the preceding results, show that n+2 n+1 < !n+1 < 1. Hence, calculate the limit of 1. In ii.) We define the sequence t, = (n + 1)/n+1!n- a.) Show the sequence (t,) is a constant sequence. b.) Deduce that for all n, tn = iii.) We define v, = nl. a.) Express vn in terms of n, tn, In and In+1. Hence, calculate the limit of v. b.) Deduce the limit of nl. %3D iv.) a.) Calculate S sin 3x dx. 22 (nl)? (2n+1)! T(2n): b.) Show that In = and that I2n+1 = 22+ (n!)?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 5 For all-natural number n, we define In = sin" x dx i.) a.) Calculate I, and !,. n+1 b.) Show that In+2 n+2 c.) Show that (I,) is a positive decreasing sequence. d.) Using the preceding results, show that n+1 < nt1 < 1. Hence, calculate the limit of n+1. In n+2 In ii.) We define the sequence t, = (n + 1)/n+1'n. a.) Show the sequence (t,) is a constant sequence. b.) Deduce that for all n, tn =. iii.) We define v, = nih. a.) Express v, in terms of n, tn, In and In+1. Hence, calculate the limit of vn. b.) Deduce the limit of ynln. %3D iv.) a.) Calculate sin 3x dx. T(2n)! 22"(nl)? b.) Show that I2n and that I2n+1 22n+ (n!) (2n+1)!
Transcribed Image Text:Exercise 5 For all-natural number n, we define In = sin" x dx i.) a.) Calculate I, and !,. n+1 b.) Show that In+2 n+2 c.) Show that (I,) is a positive decreasing sequence. d.) Using the preceding results, show that n+1 < nt1 < 1. Hence, calculate the limit of n+1. In n+2 In ii.) We define the sequence t, = (n + 1)/n+1'n. a.) Show the sequence (t,) is a constant sequence. b.) Deduce that for all n, tn =. iii.) We define v, = nih. a.) Express v, in terms of n, tn, In and In+1. Hence, calculate the limit of vn. b.) Deduce the limit of ynln. %3D iv.) a.) Calculate sin 3x dx. T(2n)! 22"(nl)? b.) Show that I2n and that I2n+1 22n+ (n!) (2n+1)!
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