4. Do the following task.a.State formally the definition of infinite seriesex an in R.C.terms of the limit of partial sum.State formally when the infinite series ¡Σª an in R converge inAm1n=1Show that the series (-1)"-1 = 1−1+1−1+1−1+...n=11diverge by showing that the limit of the partial sum s,,= 1-1+...+(-1)"-1does not exists.Hint: Consider the subsequence ($2,) and {$2n-1) of the sequence {sn}.

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Answered: 4. Do the following task. a. State…

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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The series 2 疗后店 can be shown to be conditionally convergent. Find the first 10 terms of a rearrangement of this series so that the resulting series converges to: (a) 0 (b) 1
6). State whether the series converges absolutely, conditionally, or not at all. ∞ n = 1 (−1)n + 1 1 n + 7 The series converges absolutely. The series converges conditionally. The series diverges. please show step by step clearly .
Which of the followings are convergent series? I. In2 In6 - In7 +. In3 In4 In5 II. 1(-1)" , n4+1 II. E1(-1)"" m%3D1 2n+1 3 6 2 IV. In2 7 4 + In3 :+... In7 In4 In5 In6 O a. II, II O b. III, IV O c. I, IV O d. I, I O e.Il
3) 1,2,3
Given the series s (-1)"(In(n))² n2 (a) Using the Alternating Series Error bound, determine the smallest value of N for which |s- sy < .01. (NOTE: if solve does not work, try nsolve. If that does not work, print the first several terms of the sequence |a,| and observe your value of N). (b) Use the nsum command to find Sy for your value of N in part a) and the actual sum s. (In(n))" n? (c) We now want to test for absolute convergence, namely, does con- verge. Apply the Integral Test on an appropriate function to show this series does converge. (d) Using the Integral Test Error bound, determine the smallest value of N for which |8 – SN| < .01. (Refer to the first part of the hint for part (a) here as well).
a) Determine if the geometric series below is convergent or not. 00 2" -1 Σ n =1 5h +1 The series is (Insert C if you think it is convergent and D if you think it is divergent) b) Find the sum of the series 8" -1 Σ n =1 9n +1 The sum = c) Find the partial sum S3 of the series 8-1 Σ n=1 9h +1 The partial sum S3=

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4. Do the following task. a. State formally the definition of infinite series C. terms of the limit of partial sum. n=1 State formally when the infinite series ¡Σª an in R converge in an in R. n=1 Show that the series (-1)"-¹ = 1 − 1 + 1 - 1 + 1 - 1 + ....