2. Consider the following series: (-1)n n In n (a) Is the series convergent? Justify your...
Determine whether each of the following series are divergent, condi- tionally convergent or absolutely convergent. Justify your answer. If you use a test, clearly state which test you are using. 22n +3 (b)
(-1)" (10 points each) Consider the series 회 (a) Is the series convergent? Support your answer with appropriate work and/or explanation. (b) Is the series absolutely convergent? Support your answer with appropriate work and/or explanation. (c) Is the series conditionally convergent? Explain.
1. Consider the series n=2 Is it divergent, conditionally convergent or absolutely convergent? Explain. 2. Suppose you know that 2n+1 sin(x) = Ž (-1)" 2** * Explain how to use this to show that cos(x) = ŽC-1) HINT: What is sin(x)?
1. Answer the following questions. Justify your answers. (a) (3 marks) For what values of x does the series 1 + 22x 2 + 24x 4 + 26x 6 · · · + 22nx 2n + · · · converge? (b) (9 marks) Is the following series absolutely convergent, conditionally convergent or divergent? i. X∞ n=1 2 √ n − 2 √ n + 1 ii. X∞ n=1 arctan(n) n2 + 1 iii. X∞ k=1 (−1)k k 1. Answer the...
For each of the following series decide if they are Absolutely Convergent, Conditionally Convergent or Divergent. Make sure to justify all answers. (a) º 1x5x9x.X(an+1) 3n (n!) n=0 (b) § (-1)14 n=1 (c) Î 12+3 cos(n) n n=1
(6 pts) Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series diverges Vn (n + 1)(22-1)" 2n 4 7n +4 sin(3n) (6 pts) Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series diverges Vn (n + 1)(22-1)" 2n 4 7n +4 sin(3n)
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0 Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
4. (a) Indicate where the series is (i) absolutely convergent, n-1 where it is (ii) conditionally convergent, and where it is (iii) divergent. Justify your answers Find f,(z) if f(x) = arctan (e* ) + arcsin V2x + 4. (b) (a) Set up (but do not evaluate) a definite integral that represents the area 5. of the region R inside the circle r = 4 sin θ and outside the circle r = 2. Carefully sketch the region R. (i)...
Are the following series Absolutely Convergent, Conditionally Convergent or Divergent? Explain everything. 1 24+e-n n= 1
etermine whether the following series 1. absolutely convergent /--2. divergent O3. conditionally convergent Submit answer Tn is absolutely convergent, conditionally con- vergent, or divergent. etermine whether the following series 1. absolutely convergent /--2. divergent O3. conditionally convergent Submit answer Tn is absolutely convergent, conditionally con- vergent, or divergent.