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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Random walk via Polya

A simple random walk on the integers can be described as follows . At each time unit, a walker flips a fair coin and moves one step to the right or one step to the left depending on whether the coin comes up heads or tails. Let $S_n$ denote the…
user132
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$f(\xi)=f''(\xi)$

$f(x)$ has second derivative on $[0,1]$, and $f(0)=f'(0), f(1)=f'(1)$. Prove that: there exists $\xi\in (0,1)$, s.t. $f(\xi)=f''(\xi)$. Let $g(x)=\left(f(x)\right)^2-\left(f'(x)\right)^2$, then $g(0)=g(1)=0$. By Rolle's theorem there exists $\xi\in…
Saunders
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$I_n = \int_a^b \frac{(f(x))^{n+1}}{(g(x))^n}dx$ is increasing and goes to $\infty$

Let $a$ and $b$ be real numbers with $a< b$, and let $f$ and $g$ be continuous functions from $[a, b]$ to $(0, \infty)$ such that $\int_a^b f(x) dx = \int_a^b g(x) dx$, but $f \neq g$. For every positive integer $n$, define \begin{equation} I_n =…
Math_Day
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Prove that a sequence tends to positive infinity if it is increasing and it is not bounded

I am asked to prove that $\left(a_{n}\right)_{n=0}^{\infty}$ where $a_{n+1}=\sqrt{a_{n}^{2}+a_{n}}$ tends to positive infinity as n tends to infinity.It is given that $a_{0}>0$ In order to do so,I am asked to prove it's increasing and that it is not…
Erik Dz
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construct two rational numbers

Can one explain me a bit or more about how to construct the two rational numbers? From 《Principles of Mathematical Analysis》page 2 in proving $\sqrt{2}$ is not a rational…
HyperGroups
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$e^{-nx}$ converges uniformly on $(0,\infty)$ pointwise on $[0,\infty)$

wts: $e^{-nx}$ converges uniformly on $(0,\infty)$, pointwise on $[0,\infty)$ I feel that $x \in (0,\infty)$ $\Rightarrow$ $f_n(x) \rightarrow 0$ and that $x=0 \Rightarrow f_n(x) \rightarrow 1$. When I try to prove it converges uniformly on…
ness
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Example of symbol of generalized Toeplitz operators

I am looking for the example to the sybol of generalized Toeplitz operators. For example if Toeplitz operator is matrix (S* O // 0 S) we have the symbol is matrix Y= (M_{\pH} 0 // 0 0). I do not know how we know that this Y is symbole for…
Elisabeth
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The theorem: If $x\in \mathbb{R}$ and $\forall\epsilon>0$, $|x|<\epsilon$ then $x=0$.

We have this theorem with this my proof that I will show to my students of an high school: Theorem: If $x\in \mathbb{R}$ and $\forall\epsilon>0$, $|x|<\epsilon$ then $x=0$. Proof. We supposed that $x\neq 0$. Hence $|x|>0$. If $\epsilon=|x|/2>0$,…
Sebastiano
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Find supremum/infimum and determine if max and min is assumed

What are supremum and infimum of $\{nsin\frac{1}{n}\}_{n=1}^\infty$? And is maximum and minimum assumed? I would appreciate help with the task above, I know how to find the supremum and infimum, but I do not manage to determine if the function is…
kabin
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How to show every half-open interval is a $G_\delta$ and $F_\sigma$

Every half-open interval $[a,b)$ is a $G_\alpha$ and an $F_\delta$ in $R^1$. My attempt: What I know is every $(a,b)$ is a $G_\alpha$ and an $F_\delta$ in $R^1$. Since $(a,b)= \cap (a+\frac{1}{n}, b-\frac{1}{n})$ and obviously, (a,b) is a…
Mariana
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Calculating $\lim_{x \rightarrow +\infty} \frac{e^{ax}}{x^b}$ using de l'Hospital

I am trying to calculate $\lim_{x \rightarrow +\infty} \frac{e^{ax}}{x^b}$ where a,b $>$0 Because $\lim_{x \rightarrow +\infty} x^b=+\infty$ I can use l'Hospitals Rule. So, $\lim_{x \rightarrow +\infty} \frac{e^{ax}}{x^b}=\lim_{x \rightarrow…
John.W
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How to show $\lim_{n \to \infty} \frac{1}{\sqrt{n}} \frac{1}{n} \Sigma_{k=1}^n \sqrt{k-1} = \frac{2}{3}$?

How to show $$\lim_{n \to \infty} \frac{1}{\sqrt{n}} \frac{1}{n} \sum_{k=1}^n \sqrt{k-1} = \frac{2}{3}$$ In fact, this is the lower sum of the integral of $\sqrt{x}$ from 0 to 1. So the value of the above must be $\frac{2}{3}$. But how to show?
Mariana
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Is the derivative map a surjection of differentiability classes?

It is clear to me from basic calculus that differentiability is a stronger condition than continuity, hence $C^{1}\subset C^{0}$. But, there is something that has been bothering me recently. I stumbled across the same quandary a few years back, but…
The Emerging Son
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Jensen's integral inequality conditions

In Jensen's integral inequality $\phi\bigg(\dfrac{\int_A gf}{\int_A g}\bigg)\leq \dfrac{\int_A g\,\phi(f)}{\int_A g}$, let $\phi$ be convex on $(a,b)$, which contains the range of $f$. Assume $g$ is non-negative, and $\int_A g > 0$. My question is…
Jun Xu
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Find minimum value of a

Find the minimum value of $a$ if there's a differentiable function in $\mathbb{R}$ for which : $$e^{f'(x)}= a {\frac{|(f(x))|}{|(1+f(x)^2)|}}$$ for every $x$ pretty much stuck. I think the minimum value should be $1$ but not sure.
Plom
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