# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with 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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### Prove that $\sum_{i=0}^{m_n} \Delta x_i^n\varepsilon _{i,n}(\Delta x_{i}^n)\to 0$ when $n\to \infty$.

Introduction This question is inspired from one of my other question here, and at the end, I'm really not convinced by the answer I accepted. However, I recognize that the notations are a bit confuse ...
2answers
19 views

### Study the convergence of the series

I need to study the convergence of the following series: $$\sum_{n=1}^{\infty}(\frac{1\cdot3\cdot5\cdot\cdots\cdot(2n+1)}{1\cdot4\cdot7\cdot\cdots\cdot(3n+1)}x^n)^2$$ I tried to expand that term and ...
0answers
12 views

### Unique time reachability for two symmetric ODE

We consider the following ODE $y'(t)=f(t,y(t))$ with initial conditon $y(0)=0$ and $y'(t)=-f(t,y(t))$ with initial condtion $y(0)=1$ and denote $y^+$ and $y^-$ respictevely. Assume that f is ...
0answers
18 views

### Does $\lim_{x\to 1^-}\sum_{n=0}^\infty a_n x^n\neq\infty$ implies $-\infty<\liminf\sum_{n=0}^N a_n\le\limsup\sum_{n=0}^N a_n<+\infty$?

Notation. In the text below, I consider series of real terms and use the following terminology. If $\lim_{N\to\infty}\sum_0^N a_n=\lim_{N\to\infty} s_N=\sum_0^\infty a_n=\pm\infty$ the series is ...
1answer
48 views

### Can a function from $(0,1)$ to $(0,1]$ be one-to-one and onto? [duplicate]

Does there exist a function from $(0,1)$ to $(0,1]$ both one-to-one and onto, not necessarily continuous? I couldn't think of any. Any help would be appreciated! Thanks,
2answers
34 views

### Does such a differentiable function exist?

Does there exists a differentiable function $f: \mathbb{R} \to \mathbb{R}$ with $f'(0)=0$ and the existence of a sequence $(x_n)_n$ in $\mathbb{R}$ such that $x_n \to 0$ implies $f(x_n)\to \infty$. ...
0answers
8 views

### Why does prior predictive distribution make sense?

I am trying to understand why prior predictive distribution makes sense. Let $X,\Theta,Z$ be random variables on a probability space with the probability measure $P$, and assume that $\Theta$ is ...
1answer
26 views

0answers
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### Determining the convergence/divergence of recursive sequence

Is it always possible to define a sequence using both $n^{th}$ term formula and recursion formula? For example: $a_1=3$, $a_{n+1}=a_n+3$ defines the sequence $\{3,6,9,12,...\}=\{3n\}$ I am asking ...
2answers
49 views

### Find out if $f(x,y)=\frac{1}{(1-xy)^2}$ is integrable over $[0,1]^2$

Find out if $f(x,y)=\frac{1}{(1-xy)^2}$ is integrable over $[0,1]^2$. If I could use Fubini: $$\int_0^1\int_0^1 \frac{1}{(1-xy)^2}dxdy=\int_0^1\frac{1}{1-y}dy=\infty$$\ But can't use Fubini, ...
1answer
82 views

### What is the realcompactification of the real line?

I've studied the definition of the Stone-Cech compactification by Munkres Topology. I have realized that we can't write the Stone-Cech compactification of the real line explicitly, we are just able to ...
3answers
41 views

### Orthogonal polynomials with respect to the weighting function $\omega(x)=\frac{x}{e^x-1}$

I'd like to build a family of orthogonal polynomials with respect to the weighting function $\omega(x)=\frac{x}{e^x-1}$ on $[0,+\infty)$, i.e the inner product <P_n|P_m>=\displaystyle\int_{...
0answers
31 views

### estimate $\mathcal{O}(\sqrt{n!})$ and $\mathcal{O}(\log{n!})$

How to estimate $\mathcal{O}(\sqrt{n!})$ and $\mathcal{O}(\log{n!})$. So for example $\sqrt{n!} = \sqrt{1}* \sqrt{2}* ....*\sqrt{n} \leq \sqrt{n}^n$ and $\sqrt{n!} \geq \sqrt{1}^n$. I can't find ...
1answer
15 views

### necessity of semi finiteness

As stated in this question sigma finite measures are semi finite. $\sigma$-finite measure and semi-finite measure I am interested in the question if we can weaken the sigma finiteness condition to ...
1answer
40 views

### Prove limit of the function $g = f-xf'$

Let $f:[0,\infty) \to \mathbb{R}$ be a $C^2$ function and $g :[0,\infty) \to \mathbb{R}$ be defined as $g(x) = f(x) -xf'(x)$. Prove that $f$ is convex if and only if $g$ is non increasing (and this is ...