13. Determine the convergence/divergence/absolute convergence of the series1Σ(-1)" n(Inn) 4*n=214. Determine the convergence/divergence/absolute convergence of the seriesΣ(-1)"n=21n(Inn) 3/4'Ex15. Find the sum ΣΣi=0 j=01 12i 3j+1°

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Answered: 13. Determine the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(-1)"(x+2)" The interval of convergence of the power series 2 n=0 (n+1)422n O A. -6
1 Expand the function 4+5x in a power series E anx" with center c = 0. Find a,x". n=0 (Express numbers in exact form. Use symbolic notation and fractions where needed. For alternating series, include a factor of the form (-1)" in your answer.) anx" DNE Incorrect Determine the interval of convergence. (Give your answers as intervals in the form (*, *). Use symbol ∞ for infinity, u for combining intervals, and appropriate type of parenthesis " (", ") " [" or "] " depending on whether the interval is open or closed. Enter DNE if interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) x E DNE Incorrect
1 1 Given that with center x" with convergence in (-1, 1), find the power series for 1+ 6x5 1 – x 0. TL 0 Identify its interval of convergence. The series is convergent from left end included (enter Y or N): N to x = right end included (enter Y or N): N
Σαπ M n=1 an = Σ n=1 3n+n 6"+In(n) We want to use the limit comparison test to determine whether this series converges or diverges. (a) The dominant term of the numerator is The dominant term of the denominator is We should therefore use b = We get an L= lim n+oo bn == (b) Which of the following statement is true for the series b₁₁? n=1 A. The series diverges by the nth term test for convergence. OB. The series converges by the geometric series test since -1 1. E. The series diverges by the geometric series test since r = F. none of the above >1 (c) What does the limit comparison test tell you about the series A. The test tells us no information OB. The series diverges C. The series converges D. none of the above +∞0 Σ 3" + n¹ 6n+ln(n) ? n=1
▪(– 1)"n²(x + 6)" Determine the radius of convergence of the power series 7" n=1
00 x" +1 If the series a - has the radius of 5, then the radius of convergence of the "n+1 n=1 series 2a x" is equal to .... n=1
P1 (a) Use the integral test to study the convergence of the series In(n) n² n=2 Edit: It previously said log(r) instead of ln(x). (b) Use the alternating test to study the convergence of the series 8 Σ(-1)" n=1 (c) Study the convergence of the following series n=1 2n n! (−1)n+1 2n + (−1)n Is it absolutely convergent? Remember to justify all the hypothesis needed to use the test in both parts of this exercise.
5. (a) convergence of the power series ii. Show that if lim iii. iMiMiM8 An n→∞ an+1 8 (b) Using part (a) or the original definition, find the radius of convergence of the following power series. Justify your answer. i. (n!) ² (2n)! xn log n -x”. an" where An -- {t n=0 = R exists, then R is the radius of Anxn. 1, if n k² for some ne kEN = 0, otherwise.
Let DE M₂ (R) be idempotent that is different from the identity and zero matrices. Show that the Jordan Canonical Form of D is a diagonal matrix.
2. For what values of a does the series converge? ( a) Σ ( 2)7 , α>0. (c) 2 n(in n)[In(In n))" n=1 n=3 (b) ( d) Σ(α-n)" Inn n=1 n=1

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13. Determine the convergence/divergence/absolute convergence of the series 1 Σ(-1)". n(Inn) 4* 14. Determine the convergence/divergence/absolute convergence of the series (−1)n. ∞ n=2 ∞ n=2 1 n(Inn)³/4* 15. Find the sum ∞ ∞ 1 1 ΣΣ 2i 3j+1* i=0 j=0