Prove or disprove the statement ∀n ∈ N: ln(n + 1) ≤ 1 +1/2+ · · · +1/n≤ 1 + ln(n)
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(b) Prove or disprove the statement
∀n ∈ N: ln(n + 1) ≤ 1 +1/2+ · · · +1/n≤ 1 + ln(n)
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- 1. Find the F.S. expansion of f(x) = x n = x < Xa. Prove that ƒ(x) = x - ln x is increasing for x> 1. b. Using part (a), show that ln x < x if x > 1.show that for positive epsilon, |x-4|<e/3 <=> |3x-12|<e[0 when - 7[2"e" da O 2* e* In 2 O 2e + c = +c O 2ºe² + c 1 + ln 2 ○ (2 e In 2) + cInspired by the fact that 0! = 1 and 1! = 1, let h(x) satisfy(i) h(x) = 1 for all 0 ≤ x ≤ 1, and(ii) h(x) = xh(x − 1) for all x ∈ R.(a) Find a formula for h(x) on [1, 2], [2, 3], and [n, n + 1] for arbitrary n ∈ N.(b) Now do the same for [−1, 0], [−2,−1], and [−n,−n + 1].(c) Sketch h over the domain [−4, 4]. Notice that h(x) satisfies h(n) = n! and it is at least continuous for x ≥ 0, but its piecewise definition and its many non-differentiable corners disqualify it from being our sought after factorial function. One legitimate conclusion that arises out of this exercise is that x!, when we find it, will exhibit the same asymptotic behavior as h at x = −1,−2,−3, . . . , and thus won’t be defined on the negative integers.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,