Work the problems below as directed. Show all work. Clearly mark your final answers. Use exact values unless the problem specifically directs you to round. Simplify as much as possible. Partial credit is possible, but solutions without work will not receive full credit. Part 1: These questions you will submit answers to in Canvas. Show all work and submit the work with Part 2 of the exam. But you must submit the answers in Canvas to receive credit. Each question/answer will be listed separately. The Canvas question will refer to the number/part to indicate where you should submit which answer. The questions will appear in order (in case there is an inadverten t typo). Correct answers will receive full credit with or without work in this section, but if you don’t submit work and clearly label your answers, you won’t be able to challenge any scoring decisions for making an error of any kind. 1. Find the limit of th e infinite series. (10 points) 2. Determine the convergence or divergence of each series. (6 points each) a. b. ∑ 1 (− 1)4/5 ∞ = 2 c. d. ( ) = + 1 2 4 n n n = − 1n n ne = − 1 3 1 4 1 n n = + − − 1 2 5 2 3 5 n n n n e. f. g. 3. For the series , if the series converges, does it converge conditionally or absolutely? Find the partial sum of the first six terms and then state the maximum error on the sum at that term. (16 points) 4. Find N such that for the convergent series. (10 points) 5. Use the first four terms of the Taylor series to approximate . (10 points) ( ) = + − 1 1 ln )1 ( n n n = + − 1 2 1 3 2 )1 ( n n n n n n n n 2 1 1 3 2 = + − 1 2 1 ( 1) n n n + = − 0.001 NR 4 1 1 n n = − 1 0 2dx e x 6. For the sequence below. i) Determine if the sequence is monotonic (or is monotonic after some finite value of n). You may determine this graphically or by calculating derivatives. ii) Determine the bounds (above and below of the sequence). iii) Can you apply the bounded & monotonic theorem for convergence to this sequence? iv) Does this sequence converge by another theorem? If so, which one? v) If the sequence converges, what does it converge to? ( 20 points) Part 2: In this section you will record your answers on paper along with your work. After scanning, submit them to a Canvas dropbox as directed. Th ese questions will be graded by hand. 7. Find an expression for the nth partial sum of . (12 points) 8. For each of the series in #2, state the test used to determine convergence. (9 points each) a. b. ∑ 1 (−1)4/5 ∞ =2 c. /2n na ne − = ( ) = + 1 2 4 n n n = − 1n n ne = − 1 3 1 4 1 n n d. e. f. g. 9. Determine the radius of convergence for the series . State the interval of convergence and clearly indicate whether it is open, closed or half -open. (10 points) = + − − 1 2 5 2 3 5 n n n n ( ) = + − 1 1 ln )1 ( n n n = + − 1 2 1 3 2 )1 ( n n n n n n n n 2 1 1 3 2 = + − = + + − 0 1 )1 ( )1 ( n n n x n 10. Find the nth Taylor polynomial centered at the given c. Use the included table to show work.
(15 points) n n! 0 1 2 3 4 5 6 11. Find a power series for the functions using the geometric series method. (20 points each) a. b. ( ) cos 4 , 6, 2 g x x n c = = = () ()nfx () ()nfc () n xc− () () () !
n n fc xc n − ()nPx = 3 () 21 fx x = − 3 24 7 () (1 3 ) x fx x = +
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