# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
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### Convergence radius for power series $y(x)=\sum^{\infty}_0 a_nx^n\;$ with recurrence relation [on hold]

Suppose the power series $y=\sum^{\infty}_0 a_nx^n\;$ has terms that satisfies the recurrence relation: $$a_{n+2}=\frac{(2n_1)(3n+2)}{(n+3)(2n-5)}a_n$$ with $a_0=1$,$a_1=0$. What is the radius of ...
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### Puzzled over terminology “Indefinite Integral” in Royden

In Royden (pg 125), a function of the form $$f(x)=f(a)+\int_a^x f'$$ is called an "indefinite integral" of $f'$ over $[a,b]$. However, I don't see how it is "indefinite", I thought indefinite means ...
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### Pointwise convergence: State $f(x) = \lim f_n(x)$

I'm completely confused by this subject and hoping you guys can help me to clear up my confusion. So I'm told: State $f(x) = \lim f_n(x)$ where $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ ...
Here is the problem. Let $g$ be a non-constant differentiable real function on $[a,b]$, and we know $g(a) = g(b) = 0$. I want to prove that there is a $c\in(a,b)$ such that $$\lvert g'(c)\rvert > \... 1answer 39 views ### Mean Value Theorem used to prove “global properties”? This is a soft question: I am puzzled by this statement on Wikipedia: https://en.wikipedia.org/wiki/Mean_value_theorem: The theorem is used to prove global statements about a function on an ... 2answers 31 views ### Zeros and poles of some meromorphic 1-forms on the riemann sphere Let X=\mathbb C_{\infty} be the Riemann sphere with the local coordinates \{z\ ,1/z\}. I want to show the following two statements: i) There does not exist any non-vanishing holomorphic 1-form on ... 1answer 38 views ### Integrating differential forms on curves [closed] How can I integrate the differential form$$\omega=x\,dx+y\,dy+z\,dz$$in \mathbb R^3 on the curve$$c:[0,2\pi]\to\mathbb R^3: t\mapsto (e^{t\sin t}, t^2-2\pi t, \cos \frac{t}{2})?$$Some advice ... 0answers 23 views ### A non-constant holomorphic map F between riemann-surfaces is an isomorphism I want to show the following: Let F:X\rightarrow Y be a non-constant and holomorphic map between compact riemann surfaces with genus(X)=genus(Y)\geq 2. In the above it holds that F is an ... 2answers 761 views ### Bounding \int_0^1 f(x) dx  under the condition \int_0^1 f'(x)^2 dx \le 1 Any tips on how to solve this? Problem 1.1.28 (Fa87) Let S be the set of all real C^1 functions f on [0, 1] such that f(0) = 0 and$$\int_0^1 f'(x)^2 dx \le 1 \;. $$Define ... 0answers 19 views ### Upper estimate of function such that \int |x|^2 f(x) dx<\infty Let f be a non-negative function on \mathbb R^d satisfying the following: (1) There exists a non-increasing function g on (0,\infty) such that (1-i)  C_1^{-1} g(|x|) \le f(x) \le C_1 g(|x|)... 1answer 26 views ### How do you prove triangle inequality for this metric? Let f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ be an increasing concave function such that f(t) = 0 if and only if t = 0. Let (X,d) be a metric space. Show that d_f = f \circ d  defines a ... 1answer 19 views ### Helly theorem application Let X be a normed space, dim(X)=d, let r>0, A and \subset X . Show that if every d+1 points of A are contained in a closed ball of radius r, then A is contained in a closed ball of ... 1answer 66 views +50 ### Intuition behind proof of bounded convergence theorem in Stein-Shakarchi Theorem 1.4 (Bounded convergence theorem) Suppose that \{f_n\} is a sequence of measurable functions that are all bounded by M, are supported on a set E of finite measure, and f_n(x) \to f(x) ... 0answers 20 views ### How to identify properties of the zeroes of this polynomial? [on hold] If f_0(x)=1, and f_{n+1}=\frac{d}{dx}((x^2-1)f_n(x)), prove that every f_n has exactly n distinct zeroes, all located in the interval (-1,1). It's got me stumped, so any help/pointers would ... 1answer 42 views ### f_n \to f almost everywhere and \int |f_n| \to \int |f| implies \int |f_n - f| \to 0? Suppose f_n and f are integrable, f_n \to f almost everywhere, and \int |f_n| \to \int |f|. Does it necessarily follow that$$\int |f_n - f| \to 0?
Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...