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21C Sample Final

# 21C Sample Final - Calculus Math 21C Fall 2010 Sample Final...

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Calculus : Math 21C, Fall 2010 Sample Final Questions 1. Do the following sequences { a n } converge or diverge as n → ∞ ? If a sequence converges, find its limit. Justify your answers. (a) a n = 2 n 2 + 3 n 3 2 n 3 + 3 n 2 ; (b) a n = cos( ); (c) a n = sin( n 2 ) n 2 . 2. Do the following series converge or diverge? State clearly which test you use. (a) n =1 n + 4 6 n - 17 (b) n =2 n n 4 + 7 (c) n =1 ( - 5) n +1 (2 n )! (d) n =3 ln n n (e) 1 1 4 + 1 2 4 - 1 3 4 + 1 4 4 + 1 5 4 - 1 6 4 + 1 7 4 - 1 9 4 + · · · (f) n =1 [ e n - e n +1 ] 3. Determine the interval of convergence (including the endpoints) for the following power series. State explicitly for what values of x the series con- verges absolutely, converges conditionally, and diverges. Specify the radius of convergence R and the center of the interval of convergence a . n =1 ( - 1) n 2 n n ( x - 1) n . 4. Write the Taylor polynomial P 2 ( x ) at x = 0 of order 2 for the function f ( x ) = ln(1 + x ) . 1

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Use Taylor’s theorem with remainder to give a numerical estimate of the
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21C Sample Final - Calculus Math 21C Fall 2010 Sample Final...

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