1
vote
2answers
44 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
0
votes
0answers
24 views

Riemann Integrating a Step Function

So I've been trying to prove a step function with countably infinite discontinuities is Riemann integrable using only properties of Riemann integration, no Lebesgue or gauge integration for example. ...
2
votes
2answers
47 views

Constructing $\mathbb{R}$ from $\mathbb{Q}$ and showing $\mathbb{Q}$ is dense in $\mathbb{R}$

This is a very long, multi-part problem that we were told to figure out by any means possible. There are no limits on getting help or finding answers online. I haven't had much luck at all solving ...
0
votes
1answer
24 views

Continuity of the operator

Let $D: (C^1[a,b], \Vert .\Vert_1 )\rightarrow (C[a,b], \Vert .\Vert_1)$ with $D(f)=f^{\prime}$ I wonder if I will be continuing this operator Note: $\Vert f\Vert_1=\int_a^b\vert f(x)\vert dx$ All ...
1
vote
0answers
19 views

If $f(x) = Ax$, show that for all $t \in \mathbb{R}$, the extreme $x_n = x_n(t)$ of polygon converges to $e^{At} x_0$.

Let $f$ is a vector field in $\mathbb{R}^n$, $x_0 \in \mathbb{R}^n$ and $x_{k+1} = x_k + f(x_k)\Delta t$, $k= 0,1,...,n-1$, where $\Delta t = \frac{t}{n}$. A polygon whose points are the $x_i$ ...
0
votes
0answers
16 views

Prove that $h(f(x,y))$ and $h(g(x,y))$ are first order approximations at $(x_0, y_0)$ or find a counter example

Let $f: \Bbb R^2 \to \Bbb R$ and $g: \Bbb R^2 \to \Bbb R$ be first order approximations at $(x_0, y_0)$, and let $h: \Bbb R^2 \to \Bbb R$ be continuous. Prove that $h(f(x,y))$ and $h(g(x,y))$ are ...
0
votes
2answers
37 views

Show two vectors are linearly independent

So I need help with this problem! I am confused because there is only one equation? I tried writing it in form $af(x) + bg(x) = 0$ but I really am quite stuck. Any help is greatly appreciate.
2
votes
1answer
23 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
3
votes
4answers
115 views

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent?

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ? I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other ...
1
vote
1answer
40 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
30 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
79 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
22 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
44 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
43 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
16 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
51 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
29 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
32 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
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
68 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
43 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
31 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
54 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
41 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 ...
3
votes
0answers
58 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
-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
33 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
65 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
40 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
114 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
55 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.