MATH111-200530-PS10

MATH111-200530-PS10 - MATH111-002 200530 Problem Set 10...

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Unformatted text preview: MATH111-002 200530 Problem Set 10 Edward Doolittle You do not need to hand this one in! You do not need to hand in this problem set! It is provided to help you study for the final exam. 1. (2 marks) Find the radius of convergence and interval of convergence of the following power series. ¤ ¥ ¦£ § ¤ § ¤ ¡ § ¨¤ § ¨¤ § ¨¤ § ¨¤ § (a) (b) n 0 n 1 n 0 n 1 2. (3 marks) Find the Maclaurin series for each of the following functions using the definition of Maclaurin series. 2 3. (1 mark) Sum five terms of a Maclaurin series to derive an approximation to e0 2 . Compare with the value given by your calculator. 4. (1 mark) Use a Maclaurin series derived in the textbook to obtain a Maclaurin series for the given function.  £ ¤ £  (a) f x £ sin x2 (b) f x ¤ £ xe x (compare with 2f) 5. (1 mark) Find the Taylor series for f x centered at the given value of a.  § £ § ¢ ¢ (a) f x £ 1 2x 3x2, a 2 (b) f x cos x, a π 2 6. (1 mark) Use both Maclaurin series and L’Hˆ pital’s rule to evaluate o x 0 Which do you think is easier? 7. (1 mark) Find the interval of convergence of the series    ¦¦¢ ¢ ¢ ¢ ¢ § ¤ S x § 4 6x 4x2 6x3 where cn 4 if n is even and cn of convergence. 6 if n is odd. Find a simple formula for the sum of the series on the interval Please do the following problems from the textbook. You do not need to hand in your solutions to these problems! 12.8 C-level: 3–22; B-level: 23–31; A-level: 35–40 12.9 C-level: 3–10; B-level: 13–18, 23–26; A-level: 32–40 12.10 C-level: 3–18, 47–50; B-level: 23–32, 39–42, 51–60; A-level: 61–62  lim  tan x x x3 ¥ 4x4  § ¤ £  § ©¤ £ (b) f x £ e3x (d) f x ex 1 (f) f x xe ¥ § ¤ £  ¤ ¢ £ § ©¤ (c) f x £ x  (a) f x £ cos 2x 1 1 x (e) f x ex e x ¤ ¢ £ ¤ 1 ¢ (c) 1 n (d) ¤ ¢ £ ¤ ¥ £ £ £ ¢ ∑ ∞ xn n 1 ∑ 3n ln n ∞ xn ∑ ∞ x 2 n! n ∑ ∞ 3x 1 n n 1 2n ...
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