# A06 - -1 This question should be very easy once you’ve...

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Math 138 – Fall 2011 Assignment 6 Due Monday October 31 (in the drop box by 12 noon) Hand in the following: 1. Consider the sequence a 1 = 5 , a 2 = q 5 5 , a 3 = r 5 q 5 5 , a 4 = s 5 r 5 q 5 5 , · · · etc. (a) Write this as a recursive sequence (b) Does your sequence in (a) converge? If so prove it, if not state why. 2. Consider the sequence a n = arctan( n ) (a) compute L = lim n →∞ a n (b) Given a value of > 0 find an expression for N which will guar- antee that whenever n > N, we have | arctan( n ) - L | < (c) Let = 0 . 01, using your answer from part (b), find an integer M such that | arctan( M ) - L | < BUT | arctan( M - 1) - L | (i.e. find an integer M that will guarantee that the sequence a n is within 0 . 01 of its limiting value when n M but will not be within 0 . 01 when n = M - 1. This question should be very easy

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Unformatted text preview: -1. This question should be very easy once you’ve done part b.) 3. Consider the series ∞ X n =1 1 n ( n + 1) (a) Expand the expression 1 n ( n + 1) using partial fractions and deduce a pattern for the n th partial sum s n . (HINT: Once expanded, compute a few partial sums s 1 ,s 2 ,s 3 etc. Leave each expression semi-simpliﬁed, i.e. do not ﬁnd a common denominator) (b) Compute the value of ∞ X n =1 1 n ( n + 1) (c) Find the sum of the series 1 + 1 1 + 2 + 1 1 + 2 + 3 + 1 1 + 2 + 3 + 4 + ··· 4. Show that the series ∞ X n =1 ln ± 1 + 1 n ² passes the divergence test yet fails to converge. 2...
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