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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

Real analysis is a branch of mathematical analysis, which deals with the real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

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How to express $(1+x+x^2+\cdots+x^m)^n$ as a power series?

Is it possible to express $(1+x+x^2+\cdots+x^m)^n$ as a power series?
Vafa Khalighi
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Archimedean property

I've been studying the axiomatic definition of the real numbers, and there's one thing I'm not entirely sure about. I think I've understood that the Archimedean axiom is added in order to discard ordered complete fields containing infinitesimals…
Abel
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Is $f(x)=\sin(x^2)$ periodic?

Is the function $f:\Bbb R \rightarrow \Bbb R$ defined as $f(x)=\sin(x^2)$, for all $x\in\Bbb R$, periodic? Here's my attempt to solve this: Let's assume that it is periodic. For a function to be periodic, it must satisfy $f(x)=f(T+x)$ for all…
Lazar Ljubenović
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Finding all differentiable $f: [0,+\infty) \rightarrow [0,+\infty)$ such that $f(x) = f'(x^2)$ and $f(0)=0$

After some investigation it seems fairly obvious to me that the only such function is the zero function, however I haven't been able to prove it. By considering $$\alpha =\sup\{x\in[0,+\infty) :f(x) = 0\},$$ I was able to show that $\alpha$ can…
Quantaliinuxite
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Polynomial $P(x,y)$ with $\inf_{\mathbb{R}^2} P=0$, but without any point where $P=0$

Recently I've came across such problem: give a polynomial $P(x,y)$, with $\inf_{\mathbb{R}^2} P=0$, but there is no point on the plane where $P=0$. I couldn't solve it after a day, and seriously doubt whether such a function exists, however its…
aplavin
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Arc length of the Cantor function

How does one find the arc length of the Cantor function? Wikipedia says that the length is 2. I can "see" that the length is atmost 2 by a simple trinagle inequality argument. I am struggling to come up with a partition P such that the arc length is…
Shibi Vasudevan
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n'th derivative does not vanish, but $\lim_{n\to \infty} f^{(n)}=0$.

Let $f\,$$\in$$\,C^\infty[\mathbb{R},\mathbb{R}]$ . Apparently the only functions $f$ for which there exists $n\in\mathbb{N}$ such that $f^{(n)}=0$ are polynomials in $\mathbb{R}[x]$. Is it possible to characterize the functions…
Lucien
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Monotonic version of Weierstrass approximation theorem

Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$. Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties: $p_n(x)$ is a non-decreasing function over $[0,1]$; the…
Jack D'Aurizio
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How should I prove the duality?

Rudin asked (Real Complex Analysis, First edition, Chapter 6, Problem 4): Suppose $1\le p\le \infty$, and $q$ is the exponent conjugate to $p$. Suppose $u$ is a $\sigma$-finite measure and $g$ is a measurable function such that $fg\in L^{1}(\mu)$…
Bombyx mori
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Taylor expansion for vector-valued function?

Let $f:\mathbb{R}^m \to \mathbb{R}^n$. Is it possible to do a Taylor expansion of $f$ around $\theta\in\mathbb{R}^m$? I am hoping for something like $$f\left(\theta\right) = f\left(\theta_0\right) + A \left(\theta - \theta_0\right) + \left(\theta -…
Yuki Kawabata
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Converse of the Weierstrass $M$-Test?

I was assigned a few problems in my Honors Calculus II class, and one of them was kind of interesting to do: Suppose that $f_{n}$ are nonnegative bounded functions on $A$ and let $M_{n} = \sup f_{n}$. If $\displaystyle\sum\limits_{n=1}^\infty…
Blender
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A question on convergence of series

Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$ $$ \sum_{i}a_iz_i^{n}=0 $$ does this imply that $a_i=0$…
Theo
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Is a power series uniformly convergent in its interval of convergence?

Let $R>0$ be the radius of convergence of a power series $Σa_nx^n$. Is it not uniformly convergent in $(-R,R)$? My book goes out of its way to say that if $[a,b]⊂(-R,R)$, then the power series converges uniformly in $[a,b]$. Can't we just say that…
Hrit Roy
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How to prove this integral inequality $ \int_0^{2\pi} p(x)[p(x)+p''(x)] dx \int_0^{2\pi}\frac{1}{p(x)+p''(x)} dx\geq 2\pi \int _0^{2\pi} p(x) dx $?

Let $p\in C^2(\mathbb{R})$ be a $2\pi$-periodic function such that $p(x)>0$ and $p(x)+p''(x)>0$ for all $x\in \mathbb{R}$. Then it holds $$ \int_0^{2\pi} p(x)[p(x)+p''(x)] dx \int_0^{2\pi}\frac{1}{p(x)+p''(x)} dx\geq 2\pi \int _0^{2\pi} p(x) dx…
Yuhang
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Rudin against Pugh for Textbook for First Course in Real Analysis

So as I have said before in a previous question, I am taking a first course in Mathematical Analysis, and I'm quite excited. I just found out though that unlike the other professors at my university, my professor is using Real Mathematical Analysis…
TheHopefulActuary
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