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a) If {anao and {bn}a_o are sequences of positive real numbers such that an << bn, then bn tan << bn. n=0 b) If {an}, {bn}ao and {n}o are sequences such that (vn E N)an ≤ bn ≤ en and limnoo an = limnxo Cn = LER, then limn→∞o bn = L. c) If f is a positive continuous function on [1, ∞0) such that limx→ f(x) then f (f(x))²dx converges. = 0 and f f(x)dx converges,

a) If {anao and {bn}a_o are sequences of positive real numbers such that an << bn, then bn tan << bn. n=0 b) If {an}, {bn}ao and {n}o are sequences such that (vn E N)an ≤ bn ≤ en and limnoo an = limnxo Cn = LER, then limn→∞o bn = L. c) If f is a positive continuous function on [1, ∞0) such that limx→ f(x) then f (f(x))²dx converges. = 0 and f f(x)dx converges,

Advanced Engineering Mathematics
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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Exercise 3. Determine whether each of the following statements is true or false. If one is true, provide a proof. If one is false, prove its negation, if you provide a counterexample, you need to prove that it is in fact a counterexample. a) If {a} and {n}o are sequences of positive real numbers such that an << bn, then bn tan << bn. n=0 2 b) If {an}=0, {bn}=0 and {n}a=0 are sequences such that (Vn € N)an ≤ bn ≤ cn and limn→∞an = limn→∞ Cn = L = R, then limn→∞ bn = L. c) If f is a positive continuous function on [1, ∞) such that limä→∞ f(x) then (f(x))² dx converges. = 0 and f(x) dx converges,
Transcribed Image Text:Exercise 3. Determine whether each of the following statements is true or false. If one is true, provide a proof. If one is false, prove its negation, if you provide a counterexample, you need to prove that it is in fact a counterexample. a) If {a} and {n}o are sequences of positive real numbers such that an << bn, then bn tan << bn. n=0 2 b) If {an}=0, {bn}=0 and {n}a=0 are sequences such that (Vn € N)an ≤ bn ≤ cn and limn→∞an = limn→∞ Cn = L = R, then limn→∞ bn = L. c) If f is a positive continuous function on [1, ∞) such that limä→∞ f(x) then (f(x))² dx converges. = 0 and f(x) dx converges,
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