MATH 1272 Summer 2017 Section 001 MATH 1272 Exam 3 Review Problems 1. Conceptual questions. (a) What is a sequence? Give examples. A sequence is a list of numbers in a certain order. (b) What is a series? Give examples. A sum of a list of numbers (c) What is the difference between a sequence and a series? A series involves a summation (d) What is the sequence of partial sums of a series? Pn fsng = f k=1 akg (e) What does it mean to say that a sequence converges? What is the limit of a sequence? The limit of the sequence exists. (f) What does it mean to say that a series converges? What is the value of a con- vergent infinite series? The sequence of partial sums converges. The value of a convergent infinite series is the limit of the partial sum. 1 P (g) What is the difference between saying that lim an = 10 and an = 10? n!1 n=1 1 P an = 10 means that sn ! 10 as n ! 1. n=1 2. Decide if the following statements are true or false. If true, explain your reasoning or cite necessary theorems. If false, give an example to show that the statement is false. (a) If the limit of a sequence approaches 1 , then it is equal to 1. n 1 False. a = . n 2n (b) If a sequence does not converge, then it goes to ±∞. n False. an = (−1) . (c) If the limit of a sequence is zero, then the sequence must converge. True. The definition of a convergent sequence is that it has a limit. (d) If the limit of the terms of a series is zero, then the series must converge. False. Harmonic Series. (e) If a series converges, then the limit of the terms of the series must be zero. True. This is the contrapositive of the Test for Divergence. It is also a theorem. (f) If a series diverges, the limit of the terms of the series must be non-zero. False. Harmonic Series. (g) If a series converges, then its sequence of partial sums must also converge. True. This is the definition of a convergent series. 1 MATH 1272 Summer 2017 Section 001 (h) If fang is bounded, then it converges. n False. an = (−1) . (i) If the sequence fjanjg is convergent, then the sequence fang is convergent. n False. an = (−1) 1 1 P P (j) If janj converges, then an converges. n=1 n=1 True. Absolute convergence implies convergence. 3. Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 1 4 9 16 25 ; − ; ; − ; ;::: : 2 3 4 5 6 (−1)n+1n2 a = n n + 1 4. Does the following sequence converge or diverge? If it converges, what is its limit? n sin(n)1 2 n + 1 n=1 converges to 0 1 th P 5. Assume that the n -term in the sequence of partial sums for the series an is given n=0 by 5 + 8n2 s = : n 2 − 7n2 1 P Does the series an is converge or diverge? If the series is convergent, determine n=0 the value of the series. 8 converges to − 7 −n 6. Let bn = 1 + ne . 1 (a) Does the sequence fbngn=0 converge or diverge? converges 2 MATH 1272 Summer 2017 Section 001 1 P (b) Does the series bn converge or diverge? n=0 diverges 7. Determine if the following series converge or diverge. If the series is alternating, state whether the series is absolutely or conditionally convergent. (a) 1 X 6n 5n − 1 n=0 diverges by test for divergence (b) 1 X n3 + n2 (n + 1)! n=0 convergent by ratio test (c) 1 X (−1)n+7 n2 + 3 n=0 absolutely convergent by comparison test (d) 1 X 4 n2 − 2n − 3 n=7 Convergent by comparison test (e) 1 X (−1)n−2 1 n=2 (n − 1) 3 conditionally converges by comparison test and alternating series test 3 MATH 1272 Summer 2017 Section 001 (f) 1 X n2 arctan 1 + n n=1 diverges by test for divergence (g) 1 X n2 en3 n=1 converges by integral test (h) 1 2n X 3n + 1 4 − 2n n=0 diverges by root test 8. Determine the radius and interval of convergence for the following power series. (a) 1 X (n + 1)(x − 2)n (2n + 1)! n=0 Radius is 1. Interval is (−∞; 1). (b) 1 X 6n (4x − 1)n−1 n n=0 1 5 7 Radius is . Interval is ; . 24 24 24 9. Write the function x3 f(x) = 3 − x2 as a power series. 1 X x2n+3 3n+1 n=0 4 MATH 1272 Summer 2017 Section 001 10. Give a power series representation of the derivative of the function 5x h(x) = : 1 − 3x5 1 X 5(5n + 1)3nx5n n=0 11. Find the Taylor Series for each of the following functions. (a) f(x) = ln(3 + 4x) about a = 0 1 X 4n ln(3) + (−1)n+1 xn n 3n n=1 1 (b) f(x) = about a = −3 2x 1 X 1 − (x + 3)n (6)3n n=0 5.