Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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A statement equivalent to the definition of limits at infinity?

I was fiddling around with the definition of limits at infinity and believe I have found a statement that is equivalent to the definition. So the question is this: are the following two statements equivalent? (1)…
atat
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What is the geometric intuition of $n/p$?

(volume for ball): Let B$_{r}^{p}(0):=\{x\in\mathbb R^n; \|x\|_p\leq r\}$. Then the volume of B$_{r}^{p}(0)$ is \begin{align} {\rm V}_r^{p}=2^n\cdot\frac{\left\{\Gamma(\frac{1}{p}+1)\right\}^n}{\Gamma\left(\frac{n}{p}+1\right)}\cdot r^n.…
user478705
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Proof of a binomial theorem based inequality?

Let $k \in N, x \gt 0$. Show that there exists some $n_2 \in \mathbb{N}$ so that $\forall n \geq n_2: (1+x)^n \gt n^k$. Hint: binomial theorem. My thought on this is first to make the substitution $(1+x)=b$ which means $b>1$ and $b^n>1$. This would…
John Doe
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Zeros of $ f''$

Let $ f : \mathbb{R} \to \mathbb{R} $ be a $C^2$ function such that $$ \lim_{x \to \pm \infty}{f(x)} = 0 $$ Prove that $f''$ has at least two zeros. Assume $f$ is not a constant. Than $f$ must have a stationary point, $a$. Assume it's a max point.…
astrobarrel
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Help with Dedekind cuts

I am reviewing what Dedekind cuts are for my quiz tomorrow. I had posted a question before about Dedekind cuts and I thought that was the only problem but there were these two problems as well for this unit I am having trouble understanding. I know…
MathGeek
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Application of FTC and change of variable

Let $f:[0,1]\to \mathbb{R}$ be continuous such that $$\int_{0}^{1} f(xt)dt=0$$ for all $x \in [0,1]$. Show that $f(x)=0$ for all $x \in [0,1]$. Using the FTC and substitution: $$F(t)=\frac{1}{x}\int_{0}^{t} f(u)du$$ $$F'(t)=\frac{1}{x}[f(t)]$$ I'm…
emka
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Show this sequence is equicontinuous

I'm stuck on an analysis problem to which I've reduced to the following, so some assumptions may be superfluous. Let $\{ f_n(x) \} \subset C(X,\mathbb{R}^{\geq0})$ (i.e. $f_n$ is continuous and nonnegative) where $X$ is a compact subset of…
student
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Cauchy-Schwarz for Multiple Integrals

Is there a generalization of the Cauchy-Schwarz Inequality for multiple integrals?
Shayla
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Sine not a Rational Function Spivak

This is Chapter 15 Question 31 in Spivak: a) Show sin is not a rational function. By definition of a rational function, a rational function cannot be $0$ at infinite points unless it is $0$ everywhere. Obviously, sin have infinite points that are 0…
mathnoob
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Composition of Riemann integrable functions

I know that if $f:[a,b]\to[m,M]$ is Riemann integrable and $g:[m,M]\to\mathbb{R}$ is continuous, then $g\circ f$ is also integrable on $[a,b]$. I'm trying to think about the following 3 cases: 1) $f$ the same, $g$ is Riemann integrable but not…
mez
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Show $\lim\limits_{n \to \infty} x_n=a$

Given $n,k \in \mathbb{N}$, $t_{n,k} \geq 0$, $\sum\limits_{k=1}^n t_{n,k}=1$, $\lim\limits_{n \to \infty}t_{n,k}=0$. $\lim\limits_{n \to \infty}a_n=a$ and let $x_n := \sum\limits_{k=1}^n t_{n,k}a_k$. Show $\lim\limits_{n \to \infty} x_n=a$. I…
user614287
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How to find all polynomials with rational coefficients s.t $\forall r\notin\mathbb Q :f(r)\notin\mathbb Q$

How to find all polynomials with rational coefficients$f(x)=a_nx^n+\cdots+a_1x+a_0$, $a_i\in \mathbb Q$, such that $$\forall r\in\mathbb R\setminus\mathbb Q,\quad f(r)\in\mathbb R\setminus\mathbb Q.$$ thanks in advance
M.H
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Showing a function is not Riemann integrable

Let $m,n \in \mathbb{Z_{+}}$ and let $f(x)=\begin{cases} x^m+x^n \text{ if } x \in [0,1]\cap\mathbb{Q}\\ 0 \text{ if } x \in [0,1] \setminus \mathbb{Q} \end{cases}$. I thought of a similar function $f(x)=\begin{cases} x^m \ \text{ if } x \in…
emka
  • 6,494
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Cauchy Hadamard formula and starting index of power series

The radius of convergence $r$ can be calculated for every power series $\sum_{k=0}^\infty a_k z^k$ with $a_k\in \mathbb C$ and $z\in \mathbb C$ by using the Cauchy Hadamard formula: $r = \limsup_{k\to\infty} |\frac{1}{a_k}|^{1/k}$ . In every…
Carlos
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Counter example of Lusin's theorem

The characteristic function of rationals in [0, 1] satisfies the hypothesis of Lusin's theorem. But it is no-where continuous on [0, 1]. But Lusin's theorem implies that it should be continuous on a positive measure subset of [0, 1]. What am I…
Indrani Rao Mallick
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