exam2review

# exam2review - (a f x = sin x about x = 0(b f x = xe x about...

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Exam 2 Review MAC 2312 Disclaimer: This review is by no means complete. Please study the homework as well as the lecture notes. These questions should be attempted only after you have completed all the homework assignments. You may need a calculator to solve a couple of these problems, however the exam questions will be choosen in such a way that you do not need a calculator. 1. Know the statements of ALL the theorems and how to prove the theorems given on the theorem sheet. 2. True or False. If false, give an example. (a) If X n =0 a n ( x - 3) n converges for x = 0 then the series also converges for x = 4. (b) X n =1 ( - 1) n n ! = 1 e (c) If 0 a n b n and b n diverges, then a n diverges. (d) Every function has a Taylor series expansion. 3. The Bessel function of order 0 is deﬁned by J 0 ( x ) = X n =0 ( - 1) n x 2 n 2 2 n ( n !) 2 . (a) Show that J 0 is deﬁned for all x . (b) Show that J 0 is a solution of the diﬀerential equation x 2 J 00 0 ( x ) + xJ 0 0 ( x ) + x 2 J 0 ( x ) = 0 4. Find the Taylor’s Series for the following functions

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Unformatted text preview: (a) f ( x ) = sin( x ) about x = 0 (b) f ( x ) = xe x about x = 0 (c) f ( x ) = e x about x =-1 5. Find the radius of convergence AND the interval of convergence for the following. (a) ∞ X n =2 (-1) n x n 4 n ln n (b) ∞ X n =1 (-1) n x 2 n (2 n )! (c) ∞ X n =1 ( x-5) n n ln n (d) ∞ X n =1 (-3) n ( x + 5) n √ n 6. The section 11.7 has a mix of random series. Please try them for more practice. 7. This test covers section 11.1 to section 11.10. Make sure you can tell the diﬀerence between a sequence and a series. Partial Answer sheet 2. a.) True b.) True c.) False (Why?) d.) False (Which example?) 3. a.) In class. b.) Consider section 11.9 4. a.) Each is a HW question from 11.10 5. a.) R = 4 I = (-4 , 4] b.) R = ∞ I = (-∞ , ∞ )...
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exam2review - (a f x = sin x about x = 0(b f x = xe x about...

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