Real Analysis
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Problems 15
- Is it true that for every infinite sequence of real numbers (an)n≥0(a_n)_{n\ge0}(an)n≥0, there exists a smooth function whose nnnth derivative equals ana_nan for all nnn?
- Consider a bump function ϕ :R→R\phi\colon \mathbb{R}\to \mathbb{R}ϕ:R→R, smooth with compact support. If there exists a constant MMM such that ∥ϕ(k)∥∞≤M\lVert\phi^{(k)}\rVert_\…
- Does the sequence an=1n+1n+1+⋯+12na_n = \frac1n+\frac1{n+1}+\dots+\frac1{2n}an=n1+n+11+⋯+2n1 converge to a real number? If so, evaluate the limit. If not, elaborate.
- Let f :R→Rf\colon \mathbb{R}\to \mathbb{R}f:R→R be a function, smooth at every nonzero point x≠0x\neq0x=0. Suppose f(x)=o(x3)f(x) = o(x^{3})f(x)=o(x3) near the origin, meaning…
- If A,B,CA,B,CA,B,C are positive reals such that An≥BnCA^n \ge \frac{B^n}{C}An≥CBn for n=1,2,3,…n=1,2,3,\dotsn=1,2,3,…, then what can we conclude?
- For what function f(x)f(x)f(x) is duf(u)\frac{du}{f(u)}f(u)du equal to dxf(x)\frac{dx}{f(x)}f(x)dx? \\ [Source: Communicated by Wayne Zhao; possibly due to Feynman?]
- Can we find a sequence of reals ai≠0a_i\neq0ai=0 such that each polynomial pn(x)=a0+a1x+⋯+anxnp_n(x) = a_0 + a_1 x + \dots + a_n x^npn(x)=a0+a1x+⋯+anxn has all its roots r…
- Let f :R2→Rf\colon \mathbb{R}^2\to \mathbb{R}f:R2→R be smooth, with (0,0)(0,0)(0,0) a local minimum and the unique critical point of fff. Need it be a global minimum? \\ [Sourc…
- Find ddt∫0g(t)f(x,t)dx\frac{d}{dt}\int_0^{g(t)} f(x,t)dxdtd∫0g(t)f(x,t)dx assuming fff and ggg are reasonably nice (say smooth, analytic, or even polynomial if you want). \\ […
- What is the asymptotic growth of I(n)I(n)I(n) as n→∞n\to\inftyn→∞? \\ [Source: Most recently communicated by Mark Sellke]
- Let f :R→Rf\colon \mathbb{R}\to \mathbb{R}f:R→R be an everywhere-differentiable function such that f′=ff'=ff′=f on rationals, i.e. f′(q)=f(q)f'(q) = f(q)f′(q)=f(q) for all q∈Qq…
- How could you prove this identity?
- Which of the following are ways of defining the limit superior of a sequence? Select all that apply.
- Find the cardinality of the set of continuous functions f :R→Rf \colon \mathbb{R} \to \mathbb{R}f:R→R on the real line.
- Which of the following improper integrals (some multivariable) converge? Select all that apply. [Source: Inspired by a student question during a practice lecture given in Prince…