sfinal

# sfinal - 6(20 points Give the ﬁrst four nonzero terms of...

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CALCULUS 153: SAMPLE FINAL Problem 1 (40 points) . Determine whether the following series converge or diverge. Show your work and make sure you state what test you are using. (10 points each) (a) k 2 + k + 1 2 k 3 - k + 7 (b) ( - 1) k ln k k (c) 2 k k k (d) 1 k (ln k ) Problem 2 (30 points) . Determine the convergence set for the following power series and state their radius of convergence. (10 points each) (a) 2 k x k k ! (b) ( - 1) k kx k k 2 + 1 (c) 3 k x k Problem 3 (18 points) . Solve the following differential equations. (9 points each) (a) xy = y 2 + 1 y (1) = 1 (b) xy = - 2 y + 2 e x 2 y (1) = 0 Problem 4 (36 points) . Compute the following integrals. (9 points each) (a) 1 x + 2 x ( x + 1) 2 dx (b) (ln x ) 2 dx 1

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2 CALCULUS 153: SAMPLE FINAL (c) π/ 2 0 sin 3 x dx (d) 1 0 (1 - x 2 ) - 3 / 2 dx Problem 5 (21 points) . Compute the following limits. (7 points each) (a) lim n →∞ 1 + 1 n 2 n (b) lim n →∞ n 1 / ( n +1) (c) lim x 0 + x sin
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Unformatted text preview: 6 (20 points) . Give the ﬁrst four nonzero terms of the power series ex-pansion of the following functions. Also state for which values of x it is valid. (10 points each) (a) f ( x ) = 1 2-3 x 2 (b) g ( x ) = sin( x 2 ) + xe x Problem 7 (20 points) . Let f ( x ) = e x 2-x 2-1 x 4 . (a) Write down the ﬁrst four nonzero terms of the power series expansion of f . (8 points) (b) Use the power series to compute f (4) (0), the fourth derivative of f at zero. (6 points) (c) Use the power series to ﬁnd lim x → e x 2-x 2-1 x 4 without using L’Hˆopital’s rule. (6 points) Problem 8 (15 points) . Prove that lim n →∞ x n n ! = 0 ....
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