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Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6, 8, … } is specified by the function f(n) = 2n, where n = 1, 2, 3, …. The limit of such a sequence is
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Chapter 2 Solutions
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
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- Let R be the set of all infinite sequences of real numbers, with the operations u+v=(u1,u2,u3,......)+(v1,v2,v3,......)=(u1+v1,u2+v2,u3+v3,.....) and cu=c(u1,u2,u3,......)=(cu1,cu2,cu3,......). Determine whether R is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.arrow_forwardThe Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for n=1,2,3,... a. Prove f1+f2+...+fn=fn+21 for all positive integers n. b. Use complete induction to prove that fn2n for all positive integers n. c. Use complete induction to prove that fn is given by the explicit formula fn=(1+5)n(15)n2n5 (This equation is known as Binet's formula, named after the 19th-century French mathematician Jacques Binet.)arrow_forwardIf {a} be a sequence such that lim An+¹ = 1, where | 1 | < 1, then lim an an 0.arrow_forward
- Let {an} be a monotonic sequence of real numbers. If Ban a1 = a, and an+1 = a + with a, 8 E Rt, aan +B which of the following is true for the limit of the sequence {a,}? (a) The limit is a + B. a2 +B (b) The limit is 3 (c) The limit does not exist. a + Va? + 43 (d) The limit is 2 a + Va? + B (e) The limit is 4arrow_forwardA sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence (2, 4, 6, 8, ...} is specified by the function f(n) = 2n, where n= 1, 2, 3, . The limit of such a sequence is lim f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find n00 the limit of the following sequence, or state that the limit does not exist. {-3.-1. - ; 0 } n-4 for n = 1, 2, 3, . n which is defined by f(n) = Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The limit of the sequence is 0.arrow_forwardsequence;a. Is it monotonous?b. Is it limited?c. what is the limitarrow_forward
- The limit of the sequence a.-(1+) is (a) - a)- (b) e (c) (4) G) None(e)arrow_forwardIf we know the sequence is all positive numbers, how can we prove the limit?arrow_forwardConsider the sequence a. Write out the first five terms of the sequence. a1 = a2 = a3 || a4 = a5 = b. Determine the limit of the sequence. Give an exact answer if the limit is a number. Otherwise, enter -∞ or ∞ if the limit is infinite, or enter DNE if the limit does not exist in another way. lim n→∞ 8 n + 6 n = 8 janlat - [(1 - - 46)]- 5) Ti :1 n + n=1 c. Does the sequence converge??arrow_forward
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