MAT 21C Practice Midterm Exam - 2025 PDF

MAT 21C, Practice Midterm Jan 25, 2025 Name: Student ID: Problem Points Score 1 10 2 10 3 10 4 10 5 10 6...

MAT 21C, Practice Midterm Jan 25, 2025 Name: Student ID: Problem Points Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 Total 70 1 1. Find a formula for the 0 ≤ nth term in the sequence an. State whether the sequence converges, diverges to ∞, diverges to −∞ or whether it otherwise diverges. If it converges, write the limit. (a)   −1 1 −1 1 {an } = 1, , , , ,... 4 9 16 25 (b)   8 14 20 26 {an } = , , , ,... 6 12 20 30 (c)   9 12 15 18 {an } = ,− , ,− ,... 4 6 8 10 2  ∞ X 2. Consider the following series an : n=0 n2 (a) an =. Compute the limit of the terms in the series lim an. What does the n4 + 6 n→∞ th n -term test tell you about convergence of the series?  n  n 1 1 (b) an = −3. Find a formula for the sum of the first m terms of the series. 3 4 4 (c) an = 1. Find a function f (x) for real valued x so that an = f (n) and evaluate (n + 1) 3 Z ∞ f (x)dx. What does the integral test tell you about the series? 1 3  ∞ X 4 3. Use the Limit Comparison Test to say whether the series √ converges or di- n=1 n2 +n+3 verges. 4  ∞ X 4. Consider the following alternating series (−1)n un , un ≥ 0 n=1 10 ∞ 1 X X (a) un = 6. Find a bound for an − L where L = (−1)n un. n n=1 n=1 ln(n) (b) un =. Does the series converge absolutely, conditionally or not at all? 5n 5  ∞ X 5. Consider the following power series cn (x − a)n n=1 1 (a) cn = 8 n , a = 9. Find the radius of convergence. n 8 (b) If the series converges at x = −2 and diverges at x = −15, what can you say about the radius of convergence? 6  3+x 6. Find the Taylor series generated by f (x) = 1−x at x = 2. 7  1 7. Use the fourth order Taylor polynomial for e9x at x = 0 to approximate the value of e 2. Use the Taylor series remainder estimate to bound the error in your approximation. 8