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ReviewTest2Spr08

ReviewTest2Spr08 - MAC 3473 Honors Calculus 2 Keesling...

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MAC 3473 Honors Calculus 2 Keesling Review Test #2 Test Date 3/3/08 1. Show that 1 n 3 n = 1 ! " converges. 2. Denote the limit by 1 n 3 n = 1 ! " = L . What must N be so that 1 n 3 ! L n = 1 N " < 1 10,000 ? 3. For what values of p does 1 n p n = 1 ! " converge? For what values of p does it diverge? 4. Show that 1 2 n n 3 n = 1 ! " converges using the Root Test. 5. For which values of x does x n 2 n n 3 n = 1 ! " converge? 6. For what values of x does x n n p n = 1 ! " converge for p > 1 ? For what values of x does it converge for 0 ! p ! 1 ? 7. Find a power series representation for arctan( x ) = a n x n n = 0 ! " . For which values of x does this power series converge? 8. Find a power series representation for f ( x ) = ln(1 + x ) = a n x n n = 0 ! " . For which values of x does this power series converge? 9. For which values of x do the following power series converge? (a) x n n n n = 1 ! " (b) x n n ! n = 0 ! " 10. (a) Determine the power series for sin x = a n x n n = 0 ! " . (b) What is the radius of convergence for the power series? (c) Show that the power series converges to sin x for all x for which it is defined.

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