1
vote
1answer
37 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
1
vote
1answer
17 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
6
votes
4answers
74 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
-4
votes
0answers
19 views

let E=C[X] be a normed space and T∈ L(E)… prove that.. [on hold]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
-2
votes
0answers
38 views

Let $\,E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. … [on hold]

Let $E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. Let us consider the identity $I :X→Y$. Prove that I is continuous and bijective. Calculate $\,||I||$. Prove that $I^{-1}$ is ...
2
votes
1answer
39 views

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$ $$ f(x) = \left\{ \begin{array}{c} -1, &0 \le x < 1 \\ 2, &1 \le x \le 2 ...
1
vote
1answer
12 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
0
votes
3answers
50 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
2
votes
2answers
27 views

Show that the difference quotient of $1/x^n$ exists

Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient ...
0
votes
1answer
31 views

Simple question about a complex valued function

This is taken from an exam. One and only one of the answers is true. Let $f:\mathbb R\longrightarrow\mathbb C$ such that $\lim_{x\rightarrow0}|f(x)|=+\infty$. Hence: a)There exists ...
2
votes
1answer
30 views

How to show $\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$

How to show $$\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$$ I begin with $$\sin(x+iy) = \frac{e^{x+iy}-e^{-x-iy}}{2i} = \frac{e^xe^{iy}-e^{-x}e^{-iy}}{2i}$$ $$ = ...
0
votes
0answers
12 views

How we can show that Dirichlet function is measurable with respect to borelian algebra? [closed]

Satisfying the definition of measurability? But, how we can show it?
0
votes
3answers
32 views

How to find the $n$ zeros of $\displaystyle1+z^n$?

How to find the $n$ zeros of $1+z^n$?
0
votes
2answers
24 views

How to compute $f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$

How to compute this serie : $$f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$$ The serie is convergent if $|z| < \sqrt{2} $ I can find that $$f(z) = \sum_0^{\infty} 3^{-2k}(1+i)^{-2k}z^{2k} + ...
0
votes
1answer
33 views

How to compute this integral : $\oint \bar{z}^n dz$

How to compute this integral : $$\oint_{|z|=a} \; \bar{z}\;^n dz$$ I choose $z = ae^{i \theta}$, and so $\bar{z}\;^n = a^n e^{-i\theta}$ And $$\oint_{|z|=a} \; \bar{z}\;^n dz = ...
0
votes
2answers
65 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
0
votes
0answers
25 views

Construct a Converging Series from the Following

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
0
votes
0answers
32 views

Question about deformation

Why if $H\setminus X$ has no crititical point then $X$ is a deformation retract of $H$? $H$ is a Hilbert space $X$ is a subspace of $H$ ($X=\varphi^b=\lbrace x\in H, \varphi(x)\leq b\rbrace$ , ...
0
votes
4answers
42 views

Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
-1
votes
0answers
30 views

Show $f$ is concave up if and only if graph of $f$ is above tangent line at every point

I think that this problem is intuitively obvious, and may involve Jensen's Inequality, but I am not really sure how to prove it. Any help is appreciated! Thanks
0
votes
1answer
21 views

Analysis Proof of Inflection Points

We are supposed to prove this, and it seems relatively simple, but as per usual, I don't know where to start. I assume that a big factor is that the third derivative is not zero at $x_0$, which ...
4
votes
3answers
49 views

Show a non-empty open and closed set in R must be equal to R

I did this in class, and got no credit. We are now supposed to find a proof that works, can anyone help me with this? Thanks!
1
vote
1answer
36 views

Show the image of a continuous function on a closed interval is closed.

I tried this problem on my own, but got 1 out of 5. Now we are supposed to find someone to help us. Here is what I did: Let $f:[a,b] \rightarrow \mathbb{R}$ be continuous on a closed interval $I$ ...
0
votes
0answers
16 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
-1
votes
2answers
23 views

Equation for power of a number. [closed]

Is there an equation to find power of a number?? n^m while n,m are variable I see that is hard when coming to index numbers.. so without using the log book is there any way to come up with an ...
1
vote
1answer
16 views

Relation between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$

As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$. Actually, I think that we have the inclusion ...
0
votes
1answer
37 views

Differentiable function strictly concave up $\iff f'$ strictly increasing

I feel like this is false, but I am stumped as to find a counter example. Would $f(x)=x^4$ be a candidate? Thanks!
0
votes
1answer
32 views

Show that a function $\psi : \Bbb R^n \to \Bbb R$ is affine

Fix a point x in $\Bbb R^n$. Let c be a point in $\Bbb R^n$ and define the function $\psi : \Bbb R^n \to \Bbb R$ by $$\psi(\mathbf u) = \langle \mathbf c, \mathbf u - \mathbf x \rangle \text{ for } ...
2
votes
1answer
64 views

show that if $\displaystyle\lim_{n \to \infty} f(n+x)=0$ then $\displaystyle\lim_{x \to \infty}f(x)=0$ [duplicate]

Let $f : \left[0,\infty\right]\to \mathbb R$ be uniformly continuous. If $\displaystyle\lim_{n \to \infty} f(n+x)=0$ where $x$ is in $[0,1]$ then $\displaystyle\lim_{x \to \infty}f(x)=0$ ...
0
votes
1answer
17 views

A question about $C^2$ domain.

Let $\Omega$ be a $C^2$ domain and assume that $0 \in \partial \Omega$ and that $e_n$ is orthogonal to the boundary of $\Omega$ at $0$. Then in a neighbourhood of $0$, we can put \begin{equation} ...
1
vote
2answers
38 views

Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
0
votes
0answers
21 views

$\prod_{i = 1}^n x_i^{y_i} \leq y \cdot x$

I need to prove $\prod_{i = 1}^n x_i^{y_i} \leq y \cdot x$ where $x_i \geq 0$ for all $i$ and fixed $y$ where $\sum y_i = 1$. I have looked around and all the proofs I've found have used concavity of ...
0
votes
1answer
25 views

Confusion about compact subsets of metric spaces being closed

In Rudin's Analysis, we have Theorem: Compact subsets of metric spaces are closed. Can't I generate a counterexample? $\mathbb R$ is a metric space. $(0,1)\in\mathbb R$ is a subset which is ...
2
votes
0answers
38 views

Show that a series converges

I'm very new to analysis, so this may appear quite simple. I understand intuitively why, but can't get it down formally. $$\text{Let } {x_n} \text{ be a sequence of real numbers. Suppose } x_n \to ...
1
vote
1answer
112 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
0
votes
0answers
14 views

Strictly decreasing function with a horizontal asymptote is convex?

Suppose $f$ is a strictly decreasing function with a horizontal asymptote at $t \rightarrow + \infty$. Hence, there exists a $t_{0}$ such that $\forall t>t_{0}, ~f(t)$ is a convex fuction. Is this ...
0
votes
0answers
32 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
1
vote
1answer
22 views

integration of product of even and odd function

I have a problem like this: Let $f:[-a,a]\to\mathbb R$ be a continuous function where $a>0$. If $f$ satisfies that $$\int_{-a}^a f(x)g(x)dx=0$$ for every integrable even function ...
1
vote
1answer
61 views

How to prove it?

Let $y_0\geqslant 2$, $y_n=y_{n-1}^2-2$, $n\in\mathbb{N}_+$, set $\displaystyle S_n=\sum_{k=0}^{n}\frac{1}{y_0\cdots y_k}$, how to prove $$\lim_{n\to\infty}S_n=\frac{y_0-\sqrt{y_0^2-4}}{2}.$$ Do you ...
0
votes
1answer
54 views

Prove $\nabla f(\mathbf x) = \mathbf 0.$

Suppose that the function $f:\Bbb R^n \to \Bbb R $ has first-order partial derivatives and that the point $\mathbf x$ in $\Bbb R^n$ is a local minimizer for $f:\Bbb R^n \to \Bbb R $, meaning that ...
0
votes
1answer
28 views

Fix point of a continuous function under some conditions [closed]

Prove that under each of the following conditions the continuous function $f:[a,b]\to\Bbb{R}$ has a fix point: $f([a,b])\subset [a,b]$ $f([a,b])\supset [a,b]$ When $f$ is bijective and ingective.
0
votes
0answers
16 views

Higher Order derivative of functions

can anybody help me with the solution to this question please. I am not sure how to approach the induction for this, as I don't know the significance of $k$ being even or odd,
1
vote
1answer
78 views

Gaussian Quadrature -Deriving a Formula-

eThe following is an exercise in the problem section of the Gaussian Quadrature chapter. The theorem: Derive a formula of the form $$\int_{a}^{b} f(x)dx \approx w_0f(x_0) + w_1f(x_1) + w_2f'(x_2) ...
1
vote
3answers
45 views

If Limit of function and derivative exist, then limit of derivative is 0 [duplicate]

Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0}$ but im not sure about this ...
7
votes
2answers
283 views

Real roots of a polynomial

Let $p$ be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that $p$ has at least two real roots. Thanks!
2
votes
1answer
42 views

Differentiability of a convex function

Let $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$ be convex functions such that $f\ge g$ and $f(0)=g(0)$. Show that if $f$ is differentiable in 0, then $g$ is too and $$ f'(0)=g'(0)$$ I have no idea ...
0
votes
0answers
20 views

A sequence made from a function under a certain rule.

I have been working on this question . My attemt at this question so far: $(a)$ From the diagrams I figured out the pattern to be as follows $x_{0},f(x_{0}),f(f(x_{0})),....$ wherin $x_{0}=x_{0}$ ...
0
votes
3answers
73 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
1
vote
2answers
72 views

a question about improper integral, I cannot solve it!

If $f(x)$ is continuous in $[0,\infty)$, and $\displaystyle\int_c^{\infty}\frac{f(x)}{x}\, dx$ is convergent for any $c>0$. Please prove $\displaystyle\int_0^{\infty}\frac{f(\alpha x)-f(\beta ...
1
vote
2answers
43 views

Can somebody help me solve the proof about improper integrals?

If $f(x) > 0$ is continuous at $[0, +\infty]$ and $\displaystyle \int_0^{+\infty} \frac{1}{f(x)} dx$ is convergent, please prove $\displaystyle \lim_{\lambda \to \infty} \frac{1}{\lambda} ...