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Unformatted text preview: MATH 141 – Fall 2009 – Boyle – Exam 4 Put a box around the ﬁnal answer to a question. One page per question. 1. (10 pts.) Compute the sum of the series ∞ n n=2 3(1/4) . 2. (a) (10 pts.) Find the polynomial of degree at most 15 which gives the best approximation to the function f (x) = 5 cos(x3 ) for x near zero. Answer each of (b),(c) TRUE or FALSE. No justiﬁcation needed. (b) (5 pts.) If a power series converges at x = 1, then the power series converges at x = −1. (c) (5 pts.) Every polynomial with real coeﬃcients is a product of polynomials of degree at most two with real coeﬃcients. 3. Determine whether the following series converge. Name a test which justiﬁes your answer.
∞ (a)(10 pts.)
n=2 (−1)n ln(n) (b)(10 pts.) 5n3 + 2n + cos2 (en ) √ n8 + n5 + 2 n=20
∞ 4. Write the following complex numbers z in the form a + ib with a and b real numbers. ∞ (πi/2)n 6−i (a)(7 pts.) (b)(7 pts.) (1+i)6 (c)(6 pts.) 3 + 4i n! n=0 5. Determine the radius of convergence of the following series.
∞ (a)(10 pts.)
n=1 n3x 5 n 2n 5n n (b)(10 pts.) x n=1 n!
∞ 6. (a) [6 pts.] Give a complete statement of the theorem which provides a formula for the remainder f (x) − pn (x), where pn is the nth Taylor polynomial of f about x0 . (b) [7 pts.] For the function f (x) = (x − 5)4 , compute the second Taylor polynomial p2 (x) about x0 = 3. (c) [7 pts.] Apply that remainder formula to provide an upper bound to |f (x) − p2 (x)| for x between 3 and 5, for f, p2 (x) and x0 in part (b). ...
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