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MATH 104 – Practice Midterm Exam 3 The exam will look something like this (in terms of length, type of problems etc): 1. Determine whether the integral 1 reasons clearly. 2. Let an = 3. Does 4. Does Z n+1 n Z ∞ 1 dx converges or diverges. State your + 10 x2002 1 dx. Does the sequence {an } converge? If so, what is the limit? x ∞ X n2 converge? Be sure to justify your answer. 2 n=0 (n + 4) ∞ X 10n n! n=0 converge? Be sure to justify your answer. 5. Suppose {an } is a sequence such that an 6= 0 for all n. Is it possible for both and ∞ X 1 n=1 6. Does an an n=1 to converge? Explain your answer carefully. ∞ X e2nπ n=0 ∞ X π 3ne converge? If it does, find the sum. 7. For which p does ∞ X 1 converge? p n=2 n(ln n) ∞ X 1 , by using the sum of the 4 n=1 n + 3 first ten terms. Of course, the exact error is the sum of all the terms from the 11th ∞ X 1 on, i.e., . Show that this error is less than 1/3000 by comparing this 4 n=11 n + 3 with the sum of 1/n4 and then by estimating this latter sum using an appropriate integral. 8. We would like to estimate the sum of the series 9. Use the first two non-zero terms of an appropriate series to give an approximation of Z 1 sin(x2 ) dx. 0 Give (with explanation) an estimate of the error (difference between your approximation and the actual value of the integral). 10. Find the center and radius of convergence of the power series √ ∞ X (−1)n 1 + n (x − 2)n . 2 n n=2 (In other words, find the largest open interval on which the series converges). 11. Investigate the convergence (absolute/conditional/divergence) of the series in the preceding problem at the endpoints of its interval of convergence. √ 12. Write the second-degree Taylor polynomial for f (x) = x centered at a = 25. Use √ this polynomial from to estimate 26. Also, give an estimate of the error. Here are a few extra problems to practice on: 1 Z 1. Calculate √ x ln x dx, if it converges. 0 2. Determine whether the series ∞ X n=1 √ n n n2 converges or diverges. Please explain care- fully. 3. Does the sequence {(n2 + n)1/n } converge or diverge? If it converges, calculate its limit. 4. Suppose {bn } is a sequence such that bn → B and {cn } is a sequence such that cn → C. Now let {an } be the sequence given by ( an = bn if n is even cn if n is odd Supose B 6= C. Can {an } be a convergent sequence? Explain your answer carefully. 5. Let a0 = 0, an = 6. Let an = Z 1 P . Does the series ∞ n=0 an converge? (1 + an−1 )n ∞ n 1 1+ (x2 + x4 )2 dx. Does the sequence {an } converge? If so, what is the limit? 7. Find the sum: ∞ X 2n + 1 n=1 8. Does the series 3n ∞ X n2 ln(n) n=1 n3 + 5 converge or diverge? Be sure to justify your answer. 9. Determine whether the following series converges: ∞ X ln(1 + n=3 10. Does the series ∞ X n!(n + 1)! n=1 (3n)! 1 ) n2 converge or diverge? Be sure to justify your answer. 11. For which values of p > 0 does the integral π 2 Z π 4 dx cos2 (x)(tan(x))p converge?