For posts looking for feedback or verification of a proposed solution.

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0
votes
2answers
52 views

How many permutations of the word TOMORROW can be made if the O's can't be together?

I'm trying to answer this question. This is my attempt of solution: First we distiguish the O's and R's, then we have the word: $TO_1MO_2R_1R_2O_3W$. We have $8!-7!\cdot3!-6!\cdot 3!$ different ...
0
votes
1answer
24 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
1
vote
0answers
26 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
1
vote
3answers
54 views

Right answer, wrong explanation, probability of grids?

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are ...
3
votes
4answers
83 views

$f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$

Show that $f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$ WLOG Suppose, $0< \delta \leq 1.$ Let, $\epsilon = 1$ and $x = \frac{\delta}{2}, y = x + \frac{\delta}{3}, x,y \in (0,1]$ ...
0
votes
1answer
37 views

Show that $y=\frac{4\sin\theta}{2+\cos\theta}-\theta$ is increasing function when $\theta \in [0,\frac\pi2]$

Show that $$y=\dfrac{4\sin\theta}{2+\cos\theta}-\theta$$ is increasing function when $\theta \in [0,\frac\pi2]$ What I have done If $\theta_1,\theta_2\in[0,\frac\pi2]$ then $$\sin\theta_1 < ...
1
vote
0answers
50 views

Try to solve this nested radical - I have the answer [on hold]

I have discovered the closed solution to the nested radical: $$\sqrt{n^{2^0-1}+\sqrt{n^{2^1-1}+\sqrt{n^{2^2-1}+\sqrt{n^{2^3-1}+\cdots}}}}.$$ I challenge anyone to find the closed form for any $n$. I ...
0
votes
1answer
18 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
1
vote
2answers
125 views

Find $\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$

$\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$ I let, $x = \frac{1}{n}$, then as $\lim_{x \to 0} \frac{1}{x}[(1+x)^\frac{1}{x} - e] = \infty$ L'hopital's: $\lim_{x \to 0} ...
2
votes
3answers
46 views

Uniform convergence of $f_n(x) = x^n$ on $[0,c]$

Let $c \in (0,1)$ be fixed. Let $$f_n(x) = x^n,\quad x \in [0,1)$$and$$f(x) = 0,\quad x \in [0,1)$$ Show that $f_n$ converges uniformly to $f$ on $[0,c]$. So, we have, $f_n(0) = 0, f_n( c) = ...
0
votes
1answer
13 views

solution verification: find characteristic of integral domain under given conditions

Okay, so this seems an easy problem, but I was having doubts if my solution was correct or not. I would really appreciate if somebody could verify it for me. Suppose $R$ is an integral domain such ...
1
vote
1answer
31 views

Taylor Approximation of $\cos(0.02)$

Use a Maclaurin $(a=0)$ polynomial for $\cos{(x)}$ with $3$ nonzero terms to approximate $\cos{(0.02)}$. Also, use the Taylor Remainder Theorem to find a bound on the error $\left(\displaystyle ...
3
votes
1answer
58 views

Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$, where $f(x) = 1+x^2$

Given $f(x) = 1+x^2, \alpha(x) = x^3, x \in [-1,1], P = \{-1,\frac{-1}{2},0,\frac{1}{2},1\}$. Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$. Ok, for this problem, we have $\alpha (x) = x^3$. And, I am ...
1
vote
0answers
16 views

Please could someone check and help me with my answer to part two of this exercise about vector fields along maps?

I previously solved the following (first half of an) exercise: Let $U$ be an open neighbourhood of $0 \in \mathbb R^2$ and let $f = (f_1, f_2) : U \to \mathbb R$ be a smooth map such that $f(0) = ...
0
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0answers
19 views

Please would someone check my answer to this exercise on vector fields along maps?

I believe I solved the following exercise and would appreicate it greatly if someone could check my answer: Let $U$ be an open neighbourhood of $0 \in \mathbb R^2$ and let $f = (f_1, f_2) : U \to ...
2
votes
1answer
50 views

Prove that $a_j=b_j$ for the $2$ sequences $a$ and $b$

Let $n$ is a natural number and $(n,6)=1$. Given $2$ sequences $a$ and $b$ such that $a_1>a_2>\ldots a_n$ and $b_1>b_2>\ldots b_n$. And for all $1 \leq j < k <l \leq n$, it is ...
3
votes
2answers
29 views

Show that $\frac{1}{b-a_{1}} + \frac{1}{b-a_2}+ …+ \frac{1}{b-a_n} \geq \frac{n}{b-\frac{1}{n}(a_1+a_2+…+a_n)}$

Let, $b> \max\{a_1,a_2,...,a_n\}.$ Show that $\frac{1}{b-a_{1}} + \frac{1}{b-a_2}+ ...+ \frac{1}{b-a_n} \geq \frac{n}{b-\frac{1}{n}(a_1+a_2+...+a_n)}$ f is convex if ...
1
vote
2answers
36 views

Upper and lower Riemann sums problem

Let $c > 0$ and $f(x) = x, x\in [0,c].$ Let $P = \{x_0, x_1, x_2,...,x_n\}$ be a partition of $[0,c]$ with $x_i = \frac{i}{n}c, i = 0,1,2,...,n.$ Find $U(P,f).$ Find $\lim_{n \to \infty} U(P,f).$ ...
1
vote
1answer
30 views

Evaluate products and sums

Did I evaluate the following terms correctly? Does the set notation in example b) allow me to chose the order of the terms? $$ a) \sum_{i=1}^6 ix^{i+1} = x^2+2x^3+3x^4+4x^5+5x^6+6x^7 \\ b) \prod_{i ...
2
votes
3answers
29 views

How fast is the village clock.

A man started from home at 14:30 hours and drove to a village, arriving there when the village clock indicated 15:15 hours. After staying 25 minutes, he drove back by a different route of length (5/4) ...
0
votes
1answer
13 views

Need to check an answer

At a fast food restaurant, a milk shake costs $r$. A chicken sandwich costs 3 times as much as the shake. A large order of French fries costs $\$3$. If $r-2$, how much do 3 chicken sandwiches and 2 ...
2
votes
5answers
81 views

If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then..

If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then (A) $a=c$ (B) either $a=c$ or $a+b+c+d=0$ (C) $a+b+c+d=0$ (D) $a=c$ and $b=d$ I solved $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ and got $a(a+b+d)=c(c+b+d)$ and ...
1
vote
3answers
75 views

General Solution of $\sin(mx)+\sin(nx)=0$

Problem: Find the general solution of $$\sin(mx)+\sin(nx)=0$$ My attempt: $$$$ $$\sin(mx)=-\sin(nx)$$ $$=\cos\left(\dfrac{\pi}{2}-mx\right)=\cos\left(\dfrac{\pi}{2}+nx\right)$$ Using ...
2
votes
1answer
30 views

How do I demonstrate that the given functions solve this system of ODEs?

The system is $$\left\{ \begin{array}{rcl} x'&=&y-x(x^2+y^2-1) \\ y'&=&-x-y(x^2+y^2-1), \end{array} \right.$$ and the given solution is $$x(t)=\sin(t), \quad y(t)=\cos(t) .$$ ...
1
vote
2answers
33 views

Solution verification for $y'$ when $y=\sin^{-1}(\frac{2x}{1+x^2})$

I was required to find $y'$ when $y=\sin^{-1}(\frac{2x}{1+x^2})$ This is my solution. Above when I put $\sqrt{x^4-2x+1}=\sqrt{(1-x^2)^2}$ then I get the correct answer but when I put ...
0
votes
2answers
69 views

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\sum_{n=1}^{\infty}a_n$ converges then $\sum_{n=1}^{\infty}\sin{(a_n)}$ converges.

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\displaystyle \sum_{n=1}^{\infty}a_n$ converges then $\displaystyle \sum_{n=1}^{\infty}\sin{(a_n)}$ converges. I conjecture that the final term ...
2
votes
0answers
33 views

How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ...
4
votes
1answer
63 views

Show that $f(x) = \cos(2x)$ is uniformly continuous on $[0,\infty)$

Let $f: [0,\infty) \to \mathbb{R}$ and let $f(x) = \cos(2x)$. Show that $f(x)$ is uniformly continuous on $[0,\infty)$ Mt attempt: We have, $\forall \epsilon >0, \exists \delta > 0, s.t.\mid ...
1
vote
1answer
35 views

Exercise about nbd-finiteness (Dugunji III.9.1)

Sorry for the vague title, but the question is fairly long: Let $\{A_\alpha\}$ be a ndb-finite closed cover of $X$. Consider $x_0\in X$, and let $A_{\lambda_i}$ be (all) the $A_\alpha$ that ...
0
votes
2answers
23 views

If $\inf\{diam ~(A)~|~A \in F\} =0.$ Show that $\bigcap F = \emptyset$ or $\bigcap F$ is a singleton set.

Suppose $(X,d)$ is a metric space and that $F$ is a nest of non empty subsets of $X$ for which $\inf\{diam ~(A)~|~A \in F\} =0.$ Show that $\bigcap F = \emptyset$ or $\bigcap F$ is a singleton set. ...
2
votes
1answer
31 views

What is the best time complexity of checking the inequality $a_1x_1 + \cdots + a_mx_m \le K$ to have a non-negative integer solution?

We know that all the coefficients $a_1, a_2, \ldots , a_m$ are integer. Also, $K$ is an integer number. I only need to know if the inequality has a integer solution or not. It means that there is no ...
4
votes
2answers
58 views

Show that no $T\in M_{5\times 5}(\mathbb{Q})$ has order $8$.

Before I get too far, I'll say that I think the above statement is incorrect. Assume that $T\in M_{5\times 5}(\mathbb{Q})$, with $|T|=8$, and let $f(x)=x^8-1$. Since $f(T) = 0$, it follows that, if ...
0
votes
0answers
17 views

Binomial Distribution Question… Beer Tasting Probability getting 7 or more correct

In order to select its beer tasters, a brewey give an applicant a tasting examination. The applicant is presented with five glasses, one of which contains ale and four of which contain beer, and is ...
1
vote
1answer
28 views

Suppose that X is an exponential random variable… Chebychev's Inequality

Suppose that X is an exponential random variable with pdf $f(x)=e^{-x}$ for $0<x<\infty$ and 0 otherwise. Find the exact probability that X takes on a value more than two standard deviations ...
3
votes
0answers
33 views

Estimating the sum of a series within to arbitrary certainty.

Find the sum of the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^5} = a_n$ within three decimal places. The sum is estimated by $\displaystyle a_n \approx \sum_{k=1}^{n}\frac{1}{k^5}+R(n)$ ...
3
votes
2answers
60 views

Pre-images and local homeomorphisms

I want to prove that if $f: M \to N$ is a local homeomorphism, then for all $y \in N$ we have $f^{-1}(\{y\}) \subset M$ closed and discrete. Here's the catch: this is from an exercise sheet from over ...
4
votes
4answers
104 views

Find the limit $\lim_{x\to 0}\frac{1-\cos 2x}{x^2}$

Find the limit $\lim_{x\to 0}\frac{1-\cos 2x}{x^2}$ This is what I did: $\lim_{x\to 0}\frac{1-\cos 2x}{x^2} = \frac{0}{0}$ Then, if we apply L'hopital's, we get: $\lim_{x\to 0}\frac{2\sin 2x}{2x} ...
3
votes
1answer
25 views

$n$ dimensional determinant using recurrence relations

Find determinant $$D_n(a,b,c)= \begin{vmatrix} a & b & 0 & 0 & \cdots & 0 & 0 & 0 \\ c & a & b & 0 & \cdots & 0 & 0 & 0 ...
0
votes
1answer
35 views

Number of automorphisms of $\mathbb Z _{91}$ of order 3

Solution verification, I think I have the answer worked out, just want my logic checked. Looking for the number of automorphisms of order 3 in $Aut(\mathbb Z _{91})$. So, $91=7\cdot13$, and we have ...
2
votes
1answer
25 views

$X$ connected in the order topology $\Rightarrow$ every $A\subset X$ that has an upper bound has a supremum

I have to prove $X$ connected in the order topology (w.r.t. a linear order <) $\Rightarrow$ every $A\subset X$ that has an upper bound has a supremum My attempt: Reason by contradiction: ...
4
votes
1answer
52 views

Transition probability between urn

$N$ black balls and $N$ white balls are placed in two urns so that each urn contains $N$ balls. At each step one ball is selected at random from each urn and the two balls interchange. The ...
2
votes
4answers
42 views

Separation of variables Calculus

The given differential equation I need to solve is $dy/dx=1/x$ with the initial conditon of $x=1$ and $y=10$ My attempt: $dy=\frac 1x dx$ Integrating yields \begin{align*} y&=\log x+C\\ ...
3
votes
1answer
50 views

Dependency of submatrix used in a combinatorial strategy .

This is a verification post , Please inform if anything is undefined or unclear or miss-tagged. Also if you vote up/down it would be helpfull if you leave a comment. Introduction: Given a matrix A ...
0
votes
0answers
40 views

Families of bounded (or closed) subsets of $\mathbb{R}^1$ with empty intersection, but having the finite intersection property

2.36 Theorem. If $\{K_{\alpha}\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every finite subcollection of $\{K_{\alpha}\}$ is nonempty, then $\bigcap ...
1
vote
0answers
28 views

$\alpha(t_0)$ orthogonal to $\alpha^\prime (t_0)$

Somebody asked this question a while back but only hints were posted--I want to have my entire proof examined. Let $\alpha(t)$ be a parametrized curve which does not pass through the origin. If ...
0
votes
1answer
39 views

Evaluating the convergence of the sequence $\{a_n\}=\frac{(-1)^{n-1}n}{n^2+2}$.

Set the sequence $a_n$ such that $\{a_n\}=\dfrac{(-1)^{n-1}n}{n^2+2}$. If $|a_n|$ converges (only to $0$, it would seem; correct me if I'm wrong), then $a_n$ must too converge, both to some value $L = ...
1
vote
1answer
29 views

Let X and Y be continuous random variables with joint PDF of the form $f(x,y) = c(x+y)$. Find the joint CDF

Let $X$ and $Y$ be continuous random variables with joint pdf of the form $f(x, y) = c(x+y)$ $0 < x < y < 2$ and zero otherwise. a. Find c so that f(x, y) is a joint pdf. I answered this ...
3
votes
0answers
68 views

Show that such an $f$ cannot exist

Suppose $f:\mathbb R^n\to\mathbb R$ is a scalar field, such that for a given vector $a\in\mathbb R^n$ and any $y\in\mathbb R^n-\{0\}$ we have, $f'(a;y)>0$. Show that such a function $f$ cannot ...
1
vote
1answer
31 views

Probability for rolling an odd number and tossing a coin on heads

A coin is tossed and a die rolled. Find the probability of getting a head and an odd number. The answer is $\frac{1}{4}$. My reasoning is that rolling an odd number is $\frac{1}{2}$, and tossing a ...
2
votes
1answer
37 views

Proving that $f \big|_{\partial A} = 0$, where $A = [f> 0]$.

I found this exercise: let $M$ be a metric space, and $f: M \to \Bbb R$ be a function, and $A = \{ x \in M \mid f(x) > 0\}$. Prove that if $x \in \partial A$, then $f(x) = 0$. I think that we must ...