rev1 - S (a) := X n =0 5 n ( n !) 2 (2 n !) , S (b) := X n...

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Physics (PHZ) 3113 Florida Atlantic University Mathematical Physics Fall, 2009 Review Problems I Recommended Reading: Chow, Appendix 1; Boas, Chapter 1. 1. (Chow A1.7, p. 510) Prove that sin 2 x = 1 - cos 2 x 2 and cos 2 x = 1 + cos 2 x 2 , and that A cos x + B sin x = p A 2 + B 2 sin( x + δ ) , where tan δ := A B for all real x , A and B . 2. (Chow A1.8, p. 510) Prove that cosh 2 x - sinh 2 x = 1 and sech 2 x + tanh 2 x = 1 for all real x . 3. (Chow A1.9, p. 511) Calculate the limit of the function f ( x ) := x 2 as x 2, and show that f ( x ) is continuous there. 4. (Boas 1.4.6) Evaluate the partial sums S N := N X n =1 1 n ( n + 1) , and show that they converge in the limit N → ∞ . Hint : Expand the summand using partial fractions. 5. (Boas 1.6.4) Use the comparison test to prove that series S (a) := X n =1 1 2 n + 3 n and S (b) := X n =1 1 n 2 n . converge. 6. (Boas 1.6.8 and 14) Use the integral test to determine whether the series S (a) := X n =1 n n 2 + 4 and S (b) := X n =1 1 n 2 + 9 converge or diverge.
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7. (Boas 1.6.21, 26 and 29) Use the ratio test to determine whether the series
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Unformatted text preview: S (a) := X n =0 5 n ( n !) 2 (2 n !) , S (b) := X n =0 ( n !) 3 e 3 n (3 n )! and S (c) := X n =0 p (2 n )! ( n )! converge or diverge. 8. (Boas 1.17.1, 4 and 6) Use the alternating series test to determine whether the series S (a) := X n =1 (-1) n n , S (b) := X n =1 (-3) n n ! and S (c) := X n =1 (-1) n n n + 5 converge or diverge. 9. Which of the series from the previous problem converge absolutely? 10. (Chow A1.13, p. 520) Use Gauss test to determine whether the series S := 1 2 2 + 1 3 2 4 2 + 1 3 5 2 4 6 2 + converges or diverges. Show that neither the ratio test nor Raabes test would be conclusive for this series....
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This note was uploaded on 10/21/2011 for the course PHZ 3113 taught by Professor Staff during the Fall '10 term at FAU.

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rev1 - S (a) := X n =0 5 n ( n !) 2 (2 n !) , S (b) := X n...

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