# Justify your answers state what test youre using and

• Test Prep
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Justify your answers, state what test you’re using, and remember to check any hypotheses before applying a test. (a) X n =1 2 n ( n - 1)! ( n + 1)! (b) X n =1 ( - 1) n 2 n + 7 (c) X n =1 (3 + n ) n n n +1 (d) X n =1 cos 1 n 2 7. (a) Determine all values of p R for which X n =2 1 n (ln( n )) p converges. (b) Determine if X n =2 cos( n ) sin (ln( n )) n (ln( n )) 3 converges or diverges.
8. For f ( x ) = xe x 3 , determine f (61) (0). 9. Find the radius and interval of convergence for X n =0 4 n x 2 n 10. Find the Maclaurin series for f ( x ) = - 2 x (1 + x 2 ) 2 and its radius and interval of convergence. 11. How many terms from the Maclaurin series would you need to use to approximate Z 1 / 2 0 1 + x 2 dx with error less than 15 2 13 · 4! · 9 ? 12. The Taylor Series for f ( x ) = ln 2 5 - x centred at x = 3 is X n =1 ( x - 3) n n · 2 n with radius of convergence 2. Determine the sum of X n =1 ( - 1) n n · 3 n . 13. Use series to evaluate lim x →∞ x sin 1 x . 14. Prove or disprove (that is, provide a counterexample to) each of the following statements. (a) If X n =1 a n is a positive series and if X n =1 a n converges then X n =1 a n 1 + a n converges. (b) If a n is conditionally convergent, and lim n →∞ a n a n +1 exists, then lim n →∞ a n a n +1 = 1.