Mathematics > QUESTIONS & ANSWERS > Pennsylvania State University MATH 141 CALC ANLY GEOM II In-Class Worksheet Week 12 . Solutions. W (All)

Pennsylvania State University MATH 141 CALC ANLY GEOM II In-Class Worksheet Week 12 . Solutions. Winter 2023.

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Math 141 In-Class Worksheet Week 12 Page 1 of 7 Question 1 Determine the interval of convergence of the following power series. (a) 1 X n =1 (x + 2)n n 3n (b) 1 X n =1 (-1)n (x + 2)n n 3n (c) ... 1 X n =1 (-1)n (x + 2)2n n 3n (d) 1 X n =1 (x + 2)2n n 3n (e) 1 X n =1 (x + 2)3n n 3n Question 2 (a) Suppose that the power series 1 X n =0 cn(x - 1)n converges for x = -1 and diverges for x = 5. Does the series converge for x = 2? Does the series converge for x = -5? Does the series converge for x = -2? (b) Suppose that the power series 1 X n =0 dn(x + 2)n converges conditionally for x = 8. Does the series converge for x = 7? Does the series converge for x = 9? Hint: Does the ratio test ever guarantee conditional convergence? What must be true about the ratio test limit if the series converges conditionally? Math 141 In-Class Worksheet Week 12 Page 4 of 7 rn 3 tho converges for x Interval centered here ai r interval here rn 3 tho converges for x Interval centered here ai r interval here Math 141 In-Class Worksheet Week 12 Page 5 of 7 Question 3 Suppose that f(x) = 1 X n =3 xn n (n!) and g(x) = 1 X n =3 x2n n! . (a) Calculate lim x!0 f(x) g(x) . (b) Calculate lim x!0 x4f(x) g(x) . (c) Calculate lim x!0 x3f(x) g(x) . Question 4 Find a power series representation for f(x) = 3x5 1 + 2x3 centered at 0. For which values of x is it valid? Question 5 Starting from the power series centered at 0 for g(x) = 1 1 - x , find a power series representation for arctan x centered at 0. What do we know about the radius of the interval on which this expansion is valid? You likely saw this done in class, but work through the steps to make sure you understand. [Show More]

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