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Math242E307

# Math242E307 - rules you are using(a ∞ ∑ n = 1 ln n(b...

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NAME T.A. MATH 242, Fall 2007 Exam 3: November 30 Arrange your work as clearly and neatly as possible, and cross out incorrect work. Unless otherwise noted, you must justify all answers to receive full credit. You may not use calculators, notes, or any other kinds of aids. 1. (5 points each) For the sequence ± 1 2 , 3 4 , 7 8 , 15 16 , . . . ² : (a) Find a formula for a n , the n th term of the sequence. (Assume n = 1 is the ﬁrst term.) (b) Find the limit, or show that the sequence diverges. 2. (15 points) Determine the convergence/divergence of the series n = 1 2 n + 1 + π n 3 n . State what test or rules you are using. If it converges, determine the value of the sum. 3. (15 points each) Determine the convergence/divergence of each series. State what test or
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Unformatted text preview: rules you are using. (a) ∞ ∑ n = 1 ln ( n ) (b) ∞ ∑ n = 1 ln ( n ) n 4. (15 points) Find the radius and interval of convergence for the series ∞ ∑ n = x n √ n 2 n . 5. (15 points) Find the Taylor series of sin ( 3 x ) at a = 2 π / 3. 6. (15 points) Find the ﬁrst three nonzero terms in the Maclaurin series of xe-x 1 + x . d dx ³ sin-1 x ´ = 1 √ 1-x 2 , 1-sin 2 θ = cos 2 θ , sin 2 θ = 1 2 ( 1-cos 2 θ ) d dx ³ tan-1 x ´ = 1 1 + x 2 , 1 + tan 2 θ = sec 2 θ , cos 2 θ = 1 2 ( 1 + cos 2 θ )...
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