Exercise 2

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Exercise 2. (7 points) ch Find the radius of convergence and the interval of convergence of the power series (-1) n=1 n43n Exercise 3. (4 points) Use a Riemann sum to evaluate lim =[(1 + 2) +(1+2)+...+( +--+(1+9"] + + n-00 n Exercise 4. (9 points) Consider the curve given by 2x3 + x+y - ry3 = 2. 1. Show that there are three points of e-coordinate 1 and lying on the graph of the curve. 2. Find an equation of the tangent line to the curve at each of the points found in the previous question. Exercise 5. (8 points) Determine whether the given series is convergent or divergent. n +1 Vn-1 1x 3 x 5 ... X (2n - 1) 2. en n! 1. Σ n n=1 n=1 Exercise 6. (6 points) 1. Evaluate the limit lim x2 (c/:- 1). 2. Use the Limit Comparison Test to decide whether the series Île-/– 1) converges or diverges. n=1 Exercise 7. (10 points) Compute the given integrals. 1. s tan-12 dr 22 2. | sin (2x) 1 + cos4 dc Exercise 8. (5 points) Radiaoactive substances decay at a rate proportional to the remaining mass. If a sample of a radioactive substance decayed to 94.4% of its original mass after a year, how long would it take the sample to decay to 20% of its original mass? Keep your answer as a mathematical expression. Exercise 9. (8 points) Let k be a real number. Consider the function if x 70 1-e/a f(2)= k if x = 0 1. Is there any value of k for which the function f is continuous everywhere? If yes, find that value. 2. Assume in this question that k= 0. Is f differentiable at 0? Justify your answer. Exercise 10. (8 points) 2 cos 3 - 2 cos(2.0) Evaluate the limit lim using two different methods. 2e - 2 - 23 2-0 Exercise 1. (35 points) Read each of the ten statements below carefully. • If you think the statement is true (that is, true in general), write TRUE and justify your answer, by stating a theorem, giving a proof or providing a clear reasoning. . If you think the statement is false, write FALSE and justify your answer by giving a counterexample, or providing a clear reasoning. No credit will be given to unjustified answers. Statement 1 The Ratio Test can be used to determine whether the series 1 n3 n=1 iM8 converges or diverges. Statement 2 The series Σ n n 2n +1 converges because lim n+02n + 1 1 0 and the series IM an converges, then the series (-1)"an converges. n=1 n=1 Exercise 11. (BONUS) Using series, compute lim n7 (2n)! nn Exercise 12. (BONUS) 1. Find a power series whose interval of convergence is (-3, 3). 2. Find a power series whose interval of convergence is (-3, 3).