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).

learn more… | top users | synonyms (1)

2
votes
2answers
40 views

Series of product counterexample

I get the sense that this statement is false, but I am having trouble finding a counterexample: If $\sum\limits_{n=1}^\infty a_nb_{n+1}$ converges, then $\sum\limits_{n=1}^\infty a_n b_n$ ...
2
votes
1answer
42 views

Uniqueness of the ODE solutions

Say we have a continuous function (perhaps not everywhere differentiable) that satisfies an ODE $y^\prime(x)=h(y(x),x)$ for almost all $x$ in $[0,1]$. Are the any references for that deal with basic ...
2
votes
0answers
58 views

A detail on Lusin's theorem

Suppose that $B$ is a ball of $\mathbb{R}^{m}$, $(m\geq2)$, and $f(x)$ a measurable function on $B$. According to Lusin's theorem, we can find a closed set $F\subset B$ whose complement has a measure ...
0
votes
4answers
111 views

Divergence test for $\sum_{n=1}^{\infty}\ln (1+\frac{1}{n})^n$.

I am trying to prove that this is divergent $$\sum_{n=1}^{\infty} \left(1+\dfrac{1}{n}\right)^n$$ by finding the limit of $$\ln \left(1+\dfrac{1}{n}\right)^n$$ I know its $e$ and I am trying to ...
2
votes
3answers
98 views

Proving analytical statement, Intermediate Value Theorem

Let's define $f$ as a continuous function with $f:[0;2] \to \mathbb{R}$ and $f(0) = f(2)$. Now, I want to show that: $$\exists x_0 \in [0;1]:f(x_0) = f(x_0 + 1)$$ I tried to plot a few functions in ...
0
votes
3answers
90 views

Is the series $\sum_{k=1}^\infty\frac{k}{k^2-1}$ convergent or divergent?

Is the following series convergent or divergent? And how can I prove it? Which theorem should I use? $$\sum_{k=1}^{\infty}\dfrac{k}{k^2-10}$$
2
votes
0answers
68 views

Difficulties with this integral

I have to calculate the integral $\int\sqrt{1+\cos^2x}dx$ and I do not know how to proceed! I tried to solve the differential equation associated: $y'=\sqrt{1+\cos^2x}$ then $y''=\frac{-\sin x\cos ...
4
votes
2answers
69 views

lim cos(1/θ) = 0, when θ → 0. Why?

Let $f(x)=x^2\sin\left(\frac{1}{x}\right)$ for $x\ne0$ and $f(0)=0$. => If we use Lagrange's theorem: $\exists \theta \in (0;x)$ and $f(x) - f(0) = f′(\theta)(x-0)$ => ...
0
votes
0answers
29 views

Finding a function with a given Taylor expansion

Is there any function $f(x)$ which has the following Taylor series representation? $$ f(x) = \sum_{k=0}^{\infty}{c_{k} (1 + \frac{2x^{2}}{k})^{-k/2}}. $$ for some coefficients ...
1
vote
0answers
33 views

Bivariate Correlation Unique solution

I have a set of data, let's say "score vs loss ratio", the score is an arbitrary number I gave to a team by looking at their past history. I want to prove my hypothesis which is higher the score, high ...
1
vote
2answers
29 views

Taylor theorem and a $C^{3}$ function with the following property…

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^3$ with $$f(a+h)=f(a)+f'\left(a+\dfrac{1}{2}h\right)h$$ whenever $a \in \mathbb{R}$ and $h \geq 0$. By applying Taylors Theorem to $f$ and to ...
1
vote
2answers
38 views

Prove the following statements

Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous with $f(0)=0$ and $f(1)=1$. For the following you may apply standard results without proof provided you state them carefully; $(1)$ If ...
1
vote
1answer
29 views

Existence of the compact m-cube sequence.

I have a problem connected with the proof of following lemma: If we have $Z \subset \Bbb R^n$, $m \ge 1$, $\mathcal L^m(Z)=0$ (Lebesgue measure) and $f: Z \rightarrow \Bbb R^m$ locally lipschitz, ...
0
votes
1answer
13 views

Functions with symmetrical behaviour with respect to an axis or a plane

Suppose we have two functions with a symmetrical behaviour with respect to an axis. For the sake of simplicity, let $f(x)$ and $g(x)$ have a symmetrical behaviour with respect to the $y$ axis. A ...
0
votes
2answers
32 views

Convergence of the series $\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}$.

I am trying to find out the sum (I just derived these from 2 + 0.5 + 0.125 + 0.03125 + ...): $$\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}$$ It's confusing me because it doesn't match ...
0
votes
1answer
63 views

bounded function continuous except for a set of measure zero

Let $f$ be a bounded real function on $\mathbb{R}^n$ and $P$ be a subset of $\mathbb{R}^n$ with Lebesgue measure zero. If $f$ is continuous on $P^c$, then $f$ is Riemann integrable. Is it true? my ...
5
votes
1answer
99 views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$

I'm trying hard to solve this integral but I still don't know how... $$\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$$ The integral from $-\infty$ to $\infty$ is quite easy, but how could we integrate ...
2
votes
1answer
155 views

$\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$ [duplicate]

How we can do this sum? $$ \sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90} $$ I know that we could possibly use a Fourier series decomposition however I don't know what function to start with. I ...
2
votes
1answer
61 views

How to find the closure of the following set?

How to find the closure in $l^{\infty}$ of the set $$M =\{(a_1,a_2,..........) : \textrm{all but finitely many } a_i=0\}?$$
1
vote
0answers
75 views

It does not converge, but does it diverge?

Does the sum $$\sum_{n=1}^{\infty} (-1)^n$$ diverge? It does not converge, but does it diverge?
4
votes
1answer
46 views

Find the points in the graph (my solution) - high school.

my math problem is on Swedish so i'll try my best to translate it so you can understand. I'd appreciate it if someone could point out if I did anything wrong and if there is anything that I should add ...
11
votes
4answers
431 views

Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$

Hi I am trying to find out for what values of the real parameter does the integral $$ I=\int_0^\infty \frac{\sin x}{x^s}dx $$ (a) convergent and (b) absolutely convergent. I know that the integral ...
0
votes
1answer
40 views

Maximum principle type problem

Suppose $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ is sufficiently smooth and is such that $(1/2) u_{xx} + u_{xy} + 2u_{yy} = 0$ in a ball $B$ centered at the origin. Must $u$ attain a maximum inside ...
0
votes
2answers
27 views

Closed, Compact in Metric Space

I have a concept to clarify: I understand that if we have a compact subset in a metric space, then it is closed (and bounded). If we have a closed and bounded subset of a metric space, can we also ...
0
votes
1answer
185 views

Convergence test of $\sum\limits_{n=1}^\infty ne^{-n^2}$

What convergence test should I use for $\sum\limits_{n=1}^\infty ne^{-n^2}$ ?
4
votes
1answer
72 views

finding sum of a series

I need some hint to find the sum of the series. $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n3^n}$$ I calculated using mathematica. it gives the sum as $\log(3/2)$
2
votes
1answer
49 views

Gronwall-like inequality

Let $f$ be a smooth non-negative function on $[0,\infty)$. For all $t_1\leq t_2$, consider the integral inequality $$f(t_2)+C_1\int_{t_1}^{t_2}f(s)\,\mathrm{d}s\leq C_2K(t_2-t_1)+f(t_1),$$ where ...
0
votes
3answers
52 views

Show something is a Cauchy sequence

If I want to show that something is Cauchy, should I show it converges and then show it is Cauchy or should I go at it straight from the definition. I am just trying to figure out generally what to ...
6
votes
2answers
251 views

Proof of continuity - (ε-δ) definition - Can anyone check this?

I've been trying to get my head around this problem for quite some time by now. I want to prove that $$f(x) := \left|\frac{x-1}{x^2+1}\right|$$ is continuous for $$x_0 = -1$$ Now, in order to prove ...
2
votes
2answers
55 views

properties of bounded function's second derivative

Is there a bounded function $f:\mathbb R\to\mathbb R$, such that $\forall_{t\in\mathbb R}$ $f''(t)\ne0$? If not, how to prove that?
1
vote
1answer
29 views

Cont. Function smooth iff composition with submanifold inclusion is smooth

I'm trying to proof the following: Let $X$ be a smooth manifold, $X_0$ an open subset of $X$, $i: X_0 \to X$ the canonical inclusion, $Y$ another smooth manifold and $f: Y\to X_0$ continuous, then ...
1
vote
0answers
40 views

how to show $\frac{x+x^2+\dots+x^m}{1+x+x^2+\dots+x^n}$ decreases in $x$ for $x\geq1$, where m,n are integers and 0<m<n?

I want to show $\frac{x+x^2+\dots+x^m}{1+x+x^2+\dots+x^n}$ decreases in $x$ for $x\geq 1$, assuming that $m,n$ are integers and $0<m<n$. It seems an easy question, but I can't prove it. Can ...
0
votes
1answer
23 views

Concave function first-order conditions

Let $g(x)$ be a function defined over the unit interval $[0, 1]$. Suppose that $g (0) = g (1) = 0$, and $g''(x) \le 0$, for any $x \in [0, 1]$. Show that for any $x \in (0, 1)$, $g'(x) \le ...
0
votes
1answer
31 views

Laurent series of quotient

If I have two functions $f,g$ that are holomorphic around a point $z_0 \in \mathbb{C}$. Assume the Laurent series are known and both $f$ and $g$ have a finite principal part. $$f(z) = ...
1
vote
1answer
13 views

Proof about First order derivative

Show that if $f'(c)>0$ then there exists $\delta>0$ such that $x \in (c,c+\delta) \ \ \implies \ \ f(x)>f(c)$ $x \in (c-\delta,c) \ \ \implies \ \ f(x)<f(c)$ My Attempt Now ...
1
vote
1answer
78 views

The inequality about recurrence sequence

Sequence $(x_n)$ is difined $x_1=\frac {1}{100}, x_n=-{x_{n-1}}^2+2x_{n-1}, n\ge2$ Prove that $$\sum_{n=1}^\infty [(x_{n+1}-x_n)^2+(x_{n+1}-x_n)(x_{n+2}-x_{n+1})]\lt \frac {1}{3} $$ I found relation ...
-1
votes
3answers
79 views

How many real valued Cauchy sequences are there? [closed]

Is the set of all Cauchy sequences of real numbers countable or uncountable? In other words, is $S$ countable or uncountable, where $$S=\big\{\langle x_{n}\vert ...
0
votes
0answers
12 views

Second derivative of Impulsive boundary value problem

I have this Impulsive problem : $$ \begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases} $$ and the associated functionnal is given by: ...
4
votes
2answers
82 views

Hausdorff metric and Vietoris topology

I am supposed to show that on a compact metric space, the Hausdorff metric and the Vietoris topology induce the same topology. Does anybody know how this can be done? I wanted to start by showing that ...
3
votes
2answers
343 views

Question concerning L'Hospital's rule

I know that L'Hospital's rule is applied when $\lim \frac{f'(x)}{g'(x)}$ must exist. Then, is there an example that $\lim \frac{f'(x)}{g'(x)}$ does not exist but other conditions of L'Hospital's rule ...
0
votes
0answers
231 views

Closure of a Set is Closed: Ball Proof?

I keep getting stuck with the proof that the closure of a set is closed. I'm attempting to do it by using open balls. It suffices to show that the complement of the closure of a set, say $A$, is ...
4
votes
2answers
178 views

Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$?

I wonder whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic? for every two real numbers $\alpha,\beta,\alpha\lt\beta$, $$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$ ...
1
vote
0answers
36 views

$DF(x)$ is invertible. Then is there exists $G:F(R^n) \to R^n$ such that $G(F(x))=x$??

$F:R^n \to R^n$ be continuously differentiable function. If $DF(x)$ is invertible for every $x \in R^n$. Then there exists $G:F(R^n) \to R^n$ such that $G(F(x))=x$?? Prove or disprove.
2
votes
1answer
40 views

Let $f_n: D \rightarrow \mathbb{R}: f_n(x) = g(x)^n, n≥1$. Necessary and sufficient conditions such that $f_n$ converges?

The Assignment: Let $D := [a,b]$ with $a<b$ and $g: D \rightarrow \mathbb{R}$ be continuous. Observe the sequence of functions $f_n: D \rightarrow \mathbb{R}: f_n(x) = g(x)^n, n≥1$. List and ...
4
votes
1answer
136 views

Showing $f^{(n-1)}(\xi) = 0$ for some $\xi$

Let $f$ be an $n$ times differentiable function on the interval $A$. If $x_1 < x_2 < \cdots < x_p$ are points on $A$ and $n_i, 1 \leq i \leq p,$ are natural numbers such that $n_1 + n_2 + ...
1
vote
1answer
60 views

Infinite Product Identity for Hyperbolic Sine

Prove $\prod_{n\in\mathbb{N}\backslash\left\{ 0\right\} }\left(1+\left(\frac{\alpha}{\pi n}\right)^{2}\right)=\frac{\sinh\left(\alpha\right)}{\alpha}$. I saw this formula in a book and have no idea ...
0
votes
1answer
50 views

If $a_n>0$ and $ \displaystyle \frac{a_{n+1}}{a_n} \leq 1 + \frac{1}{n^2}$ and $\displaystyle \{\frac{a_{n+1}}{a_n} \} \not\to 1$ then $a_n \to 0$

Suppose that $\{a_n\}$ is a sequence of positive real numbers and for $n \geq1$ we have: $$\frac{a_{n+1}}{a_n} \leq 1 + \frac{1}{n^2}$$ Moreover, the sequence $\displaystyle \lim_{n \to ...
9
votes
1answer
150 views

What is the radius of the largest $k$-dimensional ball that fits in an $n$ dimensional unit hypercube?

This question is adapted from another question on the 2008 Putnam test which asks specifically for the case when $n = 4$ and $k = 2$. The answer is $\dfrac{1}{2}\sqrt{\dfrac{n}{k}}$ but I am looking ...
2
votes
1answer
73 views

the series $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\sin(1+\frac{x}{n})}$ converges uniformly in $[-a,a]$

I have to show that the series $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\sin(1+\frac{x}{n})}$ converges uniformly in $[-a,a], a>0$. $$$$ That's what I have tried: ...
0
votes
1answer
33 views

pointwise and uniform convergence of the series $\sum_{n=1}^{\infty} \frac{1}{1+n^2x^2}, x>0$

I am given the exercise: Check the pointwise and uniform convergence of the series $\sum_{n=1}^{\infty} \frac{1}{1+n^2x^2}, x>0$. It is like that: $$n^2x^2 \leq 1+n^2x^2 \Rightarrow ...