Practice Final Exam - Loyola University Chicago

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5. (15 points) For what values of the parameter a, if any, does the series ∞ ∑

converge?

n=1

( a

− )

1

n+2 n+4

6. (15 points) Show that the series

converges.

∞∑

n=1

1 · 3 · 5 · · · · (2n − 1)

[2 · 4 · 6 · · · · (2n)](3 n + 1)

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Some solutions to Quiz 4 Sample A - Loyola University Chicago

5. (15 points) For what values of the parameter a, if any, does the series ∞ ∑ converge? n=1 ( a − ) 1 n+2 n+4 6. (15 points) Show that the series converges. ∞∑ n=1 1 · 3 · 5 · · · · (2n − 1) [2 · 4 · 6 · · · · (2n)](3 n + 1) 4

7. (15 points) (a) Find the radius of convergence and the interval of convergence of ∑ the power series ∞ √ x n n ; For what values of x ∈ R does the series converge 2 +3 n=1 (b) absolutely; (c) conditionally? 8. (15 points) (a) Use series to evaluate the limit lim x 2 (e −1/x2 − 1); (b) Evaluate x→∞ the nonelementary integral ∫1/64 0 tan −1 x √ x using series. 5

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Page 3: 3. (15 points) Use integration by p

Practice Final Exam - Loyola University Chicago

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