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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
77.
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Chapter 2 Solutions
Calculus: Early Transcendentals (3rd Edition)
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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_forwardsequence;a. Is it monotonous?b. Is it limited?c. what is the limitarrow_forward
- Mark all true statements (there might be more than one true statement). O Let {*} be a sequence of real numbers and S = {x: k≥ n+1} for n = N. If a = inf(S) and = sup(S), for n N then liminf(x) = lim b n n n n→ ∞ If f {xn} i is a sequence of real numbers then liminf (x)arrow_forwardLet F(n) be the function of the Fibonacci numbers, where F(n) is the nth lean of the Fibonacci numbers. The limit of F(n)/F(n+1) when n approaches infinity converges to a value of _______ a.phi -1 b. phi - 2 c. phi + 1 d. phi+2arrow_forwardarrow_back_iosarrow_forward_ios
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