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3
votes
2answers
42 views

Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$$ Suppose ...
1
vote
0answers
39 views

Convergence of $\sum_{n=3}^{\infty}\frac{1}{n\log n(\log\log n)^\alpha} $

Does the following series converge: $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n\log n(\log\log n)^\alpha} $ and $\alpha>0$ ? Using Cauchy condensation test twice: $\begin{align} ...
0
votes
1answer
9 views

$\frac{G_1\times G_2}{N_1\times N_2}\cong \bigg(\frac{G_1}{N_1}\bigg)\times \bigg(\frac{G_2}{N_2}\bigg)$

Is my solution to this question correct? If $N_1\triangleleft G_1,N_2\triangleleft G_2$, then $(N_1\times > N_2)\triangleleft (G_1\times G_2)$ and $\frac{G_1\times G_2}{N_1\times N_2}\cong ...
2
votes
2answers
44 views

If $\sin A = \cfrac{3}{5}$ with $A$ in QII, find $\sec2A$.

If $\sin A = \cfrac{3}{5}$ with $A$ in QII, find $\sec2A$. I'm getting $\sec2A=\cfrac{25}{7}$. Is that correct?
0
votes
2answers
42 views

If $\sin (B) = − \frac 1 2 $ with $B$ in third quadrant, then find $\cot (B/2)$

If $\sin (B) = − \frac 1 2 $ with $B$ in third quadrant, then find $\cot (B/2)$ I'm getting $-\sqrt{3}-2$
0
votes
1answer
20 views

If $\sin A = 4/5$ with $A$ in QII, find $\cos A/2$

For the following, assume that all the given angles are in simplest form, so that if A is in QIV you may assume that 270° < A < 360°. If $\sin A = 4/5$ with A in QII, find $\cos A/2$ I keep ...
0
votes
2answers
30 views

If $\sin B = −1/2$ with $B$ in QIII, find $\cos B/2$

For the following, assume that all the given angles are in simplest form, so that if A is in QIV you may assume that 270° < A < 360°. If $\sin B = −1/2$ with B in QIII, find $\cos B/2$ Here's ...
1
vote
1answer
50 views

Prove $\cos 3\theta = 4 \cos^3\theta − 3 \cos \theta$

$\cos 3θ = 4 \cos^3 θ − 3 \cos θ$ Here's my attempt. Is it correct? Thanks! $\cos(3θ)$ $= \cos(2θ + θ)$ $= \cos(2θ)\cos(θ) - \sin(2θ)\sinθ$ $= (2\cos^2θ - 1)\cosθ - (2\sinθ\cosθ)\sinθ$ $= ...
0
votes
1answer
16 views

System of inequalities. Points of intersection?

$x^2+y^2<=81$ $y<x$ Is this correct? My answer: (-9sqrt(2)/2,-9sqrt(2)/2), (9sqrt(2)/2,9sqrt(2)/2)
1
vote
1answer
20 views

Check my solution to this trig inequality

Problem $1.88$ : Solve $$\cos x\lt \frac{\sqrt{3}}{2},\qquad x \in [0,2\pi]$$ I found the set of solutions to be $S=[0,2\pi]-\left[\dfrac{\pi}{6},\dfrac{11\pi}{6}\right]$ Is this correct? Thank you.
0
votes
2answers
40 views

Feedback on Euclidean Algorithm: $gcd(277, 301)$

Ans: $301 =277 \cdot 1 + 24$ $277 =24 \cdot 11 + 13$ $24 = 13 \cdot 1 + 11$ $13 = 11 \cdot 1 + 2$ $11 = 2 \cdot 5 + 1$ $2 = 1 \cdot 2 + 0$ Is this correct?
0
votes
1answer
30 views

Radius of Convergence of $\sum_{n=0}^{\infty}n^2z^n$

Find the radius of convergence of $$\sum_{n=0}^{\infty}n^2z^n$$ (If it matters, $z$ is a complex variable.) My attempt: The radius of convergence is \begin{align*} ...
2
votes
0answers
14 views

action of symmetry group of cube on pairs of opposite faces.

I want to solve the following problem from Dummit & Foote's Abstract Algebra: Explain why the action of the group of rigid motions of a cube on the set of three pairs of opposite faces is not ...
0
votes
0answers
18 views

Identifying the abelian group with a presentation matrix

I am doing problems from Artin: \begin{bmatrix} 2 \\ 1\\ \end{bmatrix} and \begin{bmatrix} 2 & 4\\ 1 & 4\\ \end{bmatrix} For the First one after manipulating rows I ...
0
votes
0answers
4 views

Mirror a line over a plane

I am trying to mirror a line over a plane, but I am not sure if I am doing it right, so please tell me if something that I do is wrong. I have 2 points $A(1, 2, 1);B(-1,0,2)$ and I have to mirror the ...
0
votes
2answers
34 views

probability: arithmetic with random variables.

I have a question of using arithmetic on random variables. Please refer to the following question, to which I will present my solution using the arithmetic (which I thought it's correct but actually ...
0
votes
0answers
40 views

Solutions of Ax=b

I'm studying about linear equations and I've stumbled upon a problem I have an equation of the type $Ax = b$ where $A$ is a matrix, $x$ is a vector of unknown variables, and $b$ is a vector of the ...
1
vote
0answers
25 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
0
votes
1answer
43 views

Parabola problem. How far from the vertex is the focus?

A spotlight has a parabolic cross section that is 8 ft wide at the opening and 2.5 ft deep at the vertex. How far from the vertex is the focus? I'm getting 8/5 ft. Is this correct? Appreciated!
0
votes
2answers
45 views

Is this integral wrong?

$$ \int(x^2+1)(x+1)dx$$ Per partes, let $$u = x+1; u' = 1 $$ $$v' = x^2+1; v = x^3/3 + x $$ $$=(x+1)(x^3/3+x) - \int1*(x^3/3+x) = (x+1)(x^3/3+x) - \frac{1}{3}\frac{x^4}{4}-\frac{x^2}{2} + C $$ ...
0
votes
0answers
15 views

Is this a Partial Order relation?

Doing some review for my exam. Here is a sample question: Find the partial order relation on {1,2,3} that contains (1,2) and (2,3). My attempt: R: { (1,2), (2,3), (1,3), (1,1), (2,2), (3,3) } ...
0
votes
0answers
20 views

Coefficient of Determination (Alternative Solution)

Consider the following problem - where I have purposefully omitted the numbers in question since it is of no interest for my question. From $40$ observations on \begin{align} ...
0
votes
0answers
19 views

Find the general solution to diophantine equation $-221x + 187y - 493 = 0$

I have to find the general solution to $$-221x + 187y - 493 = 0$$ The main issue, I'm figuring out if I have found the general solution or not. Below, are my steps: The $\gcd{(-221,187)} = 17$ and ...
0
votes
1answer
30 views

Dice - conditonal probability

I just need quick solution check. 100 dice are rolled. What is probability of getting die with number 5 at least twice if there is exactly 20 dice with number 6? My solution: Let $A=${at least ...
1
vote
2answers
36 views

Parabola problem involving depth.

The equation for the parabola, I found, is $$y^2=3.2x$$ What is the depth of the reflector? I'm getting $\dfrac {25}{3.2}$.
-1
votes
0answers
40 views

Parabola word problem. Depth? [duplicate]

I did the first part of this problem correctly, coming up with the equation of y^2=3.2x However, for the second part, where it wants to know the depth of the reflector, I’m having trouble. I ...
0
votes
1answer
9 views

Did I correctly calculate the specificity and the false negative rate?

So I filled out the summary table of the data, but I'm not quite sure if I calculated the specificity and the false negative rate correctly from the table. Can someone please check that I'm doing it ...
1
vote
1answer
44 views

How do I find the probability of specified events from a permutation of the 26 english letters?

I found a similar problem here, but I don't really understand the explanation to their solution and can't apply it. Question: What is the probability of the following even when we randomly select a ...
0
votes
0answers
16 views

Am I calculating probability correctly?

I am working on some discrete math problems and would like to make sure I'm doing them correctly and what not. Instead of typing it all out in latex, I took a picture of the questions/my work. I hope ...
0
votes
1answer
38 views

Find a vector parallel to the line of intersection of the two given planes.

The given planes are $2x - y + z = 1$ and $3x + y + z = 2$. Adding the two equations I choose the two points of intersection of the planes to be $P= (3, -1, -6)$ and $X = (5, -2, -11)$. Then, the ...
0
votes
1answer
24 views

Word Problem — System of Equations

I think I'm doing this wrong. Can anyone help me identify the problem? Thanks. A hardware supplier manufactures three kinds of clamps, types A, B, and C. Production restrictions force it to make 10 ...
1
vote
2answers
78 views

drop independent random variables in conditional expectation expression

Is the following true for discrete random variables $K,Y,X_1,...$ where $K$ is independent of $\sigma(Y,X_1,...)$? $$E[\sum_{i=1}^K X_i | K,Y] = \sum_{i=1}^K E[X_i|K,Y] = \sum_{i=1}^K E[X_i|Y]$$ ...
3
votes
0answers
26 views

normal of a function

Function $$\mathbf{r}(u,v)=(2u+2v)\mathbf{i}+(-3+v^2)\mathbf{j}+(2u^2)\mathbf{k}$$ is a parametrization of a surface. What's the normal vector of this surface at the point $(u,v)=(1,-2)$. What I got: ...
1
vote
1answer
30 views

Differential equation problem!

I've found an answer to a question, which is apparently incorrect but I have no idea why. "Find the solution of the differential equation that satisfies the given initial condition." ...
5
votes
2answers
73 views

Show that $||x| - |y|| \le |x-y|$ for all $x, y, \in \mathbb{R}$

Show that $||x| - |y|| \le |x-y|$ for all $x, y, \in \mathbb{R}$ Working: So we know that $|x| = |(x-y) + y | \le |x-y|+|y|$ Thus, $|x|-|y| \le |x-y| \ \cdots (1)$ and similarly, $|y|-|x| = ...
3
votes
1answer
54 views

Are these Laplace transforms wrong in Stroud's Advanced Engineering Math Book?

I know that if you think a book is wrong, most probably it is your own mistake. However, I can't understand the following Laplace transforms in K. A. Stroud's "Advanced Engineering Mathematics". In ...
1
vote
1answer
117 views

Better way to prove this sequence problem relating $\lim x_{n+1}/x_n$ and $\lim\sqrt[n]{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis. Let $X= (x_n)$ be a sequence of strictly positive real numbers, let $\lim \left({ \dfrac{x_{n + 1}}{x_n}}\right) = L$, ...
15
votes
4answers
1k views

How many solutions for this equation?

$$ \frac{x-4}{(x-1)} = \frac{1-4}{(x-1)} $$ Can someone tell me how many solutions are there for the above equation? MY APPROACH: I cross multiplied the equations and re-arranged to get a quadratic ...
0
votes
0answers
33 views

Green function and Fourier series

I would like to understand what is wrong with the following computation. Indeed the result appears rather trivial. I have the following differential equation $$ y''(t,t')+z(t)y(t,t')=\delta(t-t') ...
0
votes
2answers
34 views

Placing black and white balls in a row, s.t. no pair of black balls is lying side by side

In how many ways can we dispose $m$ white and $n$ black balls such that there is no pair of black balls lying side by side ? If $n>m+1$ it is not possible, if not; I place the black balls, ...
0
votes
0answers
14 views

Implicit and Explicit formulation of this PDE

I am trying to solve the following equation by using both implicit as well as explicit formulation. $$\frac{\partial^2Q_i}{\partial t^2} = ks_{ij} \left( \frac{Q_j}{Ad_j} - \frac{Q_i}{Ad_i} \right) ...
2
votes
1answer
54 views

Sum of sawtooth function not differentiable at dyadic rational points

This question is related to this question. Define $h(x)=|x|$ on $[-1,1]$ and extend it to $\mathbb R$ by defining $h(2+x) = h(x)$. This is a sawtooth function that is $0$ at even and $1$ at odd ...
0
votes
1answer
51 views

Exponential Growth Problem. Is this solution correct?

I would just like to confirm that I'm doing this correctly. If not, any help would be appreciated. Thanks! The problem: A painting sold for $$274 in 1977 and was sold again in 1987 for $470. Assume ...
2
votes
1answer
27 views

Please can you check my proof of $f$ is Lipschitz

I tried to prove: If $f$ is differentiable on $[a,b]$ and if $f'$ is continuous on $[a,b]$ then $f$ is Lipschitz continuous on $[a,b]$. Please can you tell me if my proof is correct: Proof: Since ...
1
vote
1answer
41 views

Please can you check my computation of the limit of this sum of sawtooth function

Define $h(x)=|x|$ on $[-1,1]$ and extend it to $\mathbb R$ by defining $h(2+x) = h(x)$. This is a sawtooth function that is $0$ at even and $1$ at odd integers. Furthermore define $h_n(x) = (1/2)^n ...
0
votes
0answers
45 views

Condition of a matrix proof

I have to show the following inequality: For given two invertible matrices $A,B \in \mathbb{R}^{n\times n}$ show that $k(AB)\leq k(A)k(B)$, where $k(A)=\left \| A \right \|\left \| A^{-1} \right \|$ ...
3
votes
1answer
52 views

Checking if $\langle 2 \rangle$ is a maximal ideal in $\mathbb{Z}[i]$

Is $\langle 2\rangle$ a maximal ideal in $\mathbb Z[i]$? Solution: We know that $\mathbb Z[i]$ is ED And hence PID. Consider $2\in\mathbb Z[i]$. Then $N(2)=2^2=4$ (NOTE: $N$ is norm). Since $N(2)$ is ...
0
votes
1answer
29 views

Uniform convergence and Integrals

For $x \in \mathbb{R}$ and $n \in \mathbb{N}$, let $f_n(x) = \frac{x}{1+nx^2}$ and $f(x)=0$. (i) Prove $f_n \to f$ uniformly. (ii) Calculate $\displaystyle{\lim_{n\to\infty}\int_0^1} f_n(x)dx$ in ...
7
votes
3answers
428 views

Sequences of Rationals and Irrationals

Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational? Let $(y_n)$ be a sequences that converges to the rational number ...
1
vote
2answers
50 views

Solution Verification: Convergence of Norms $\implies$ Convergence of Sequence

I found the following exercise in Bartle's Elements of Real Analysis. I'm learning on my own and have a couple of doubts would love it if someone could take a look. ($\left|{\left|{x}\right|}\right|$ ...