Rational Numbers & Definition of Convergence of A Sequence Applied Analysis Problems

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7 problems altogether. Clear hand written work would be good enough. Thank you!!

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(2) For any x, y ∈ R with x < y, show that there are rational numbers r, q such that √ x < r 3 + q < y. (3) Use the ε − N definition of convergence of a sequence to verify the limit, √ √ √ 3 + 2n2 2 √ lim √ . = n→∞ 3 2n + 3n2 (4) Let {an } be a sequence of positive numbers. Define the sequence {bn } by √ sin an + a2n + 1 bn = . an + 1 Show that {bn } has a convergent subsequence. (5) The sequence {an } is defined by 3n − 1 2 nπ 1 an = sin ( + ). n+1 6 n Find the upper and lower limits of the sequence and all the limit points of {an ∶ n = 1, 2, . . .}. (6) The sequence {an } is defined by a1 = 1 and 1 1 1 an+1 = (2an + 1) 2 − (2an + 1) 4 , n = 1, 2, . . . 4 Show that the sequence is convergent. √ (7) Suppose that lim nan ≥ 1. Test the following series for convergence, n→∞ ∞ an + √1n n=1 n2 a2n ∑ . (8) Show that the following series is conditionally convergent, ∞ sin nπ 7 . ∑ 1 1 √ n=2 ln (1 + 2 ) + ⋯ + ln (1 + √n ) Purchase answer to see full attachment Tags: rational numbers positive numbers definition of convergence lower limits conditionally convergent User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.