qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,179,498 | <p>Is the following sequence convergent in <img src="https://i.stack.imgur.com/wtYTG.png" alt="enter image description here">?(Metric space is the normal, euclidean space)
<img src="https://i.stack.imgur.com/uqGFl.png" alt="enter image description here"></p>
| Michael Hardy | 11,667 | <p>Quid's answer is in one sense the same as what I was about to post when I read it. But the way I phrased it may make it clear to some people first learning the subject, in a way that quid's might not.</p>
<p>Suppose an isomorphism $\varphi$ from the multiplicative group to the additive group exists. In a field in... |
2,471,217 | <p>Contradiction or contra-positive? Or is direct easier? </p>
<p>∀n∈N, (n>3∧ n is prime)→∃q∈N,(n=6q+1∨n=6q+5)</p>
| Peter | 82,961 | <p>In plain text the expression means : Every prime $n$ greater than $3$ has the form $6q+1$ or $6q+5$, which is clearly the case because the numbers $6q,6q+2,6q+3,6q+4$ are divisible by $2$ or $3$.</p>
|
3,164,343 | <p>Machine generates one random integer in range <span class="math-container">${[0;40)}$</span> on every spin.<br>
You should choose 5 numbers in that range. </p>
<p>Then the machine will spit out <strong>5</strong> numbers (<em>numbers are independent of each other</em>). </p>
<p><strong>what is the probability t... | lulu | 252,071 | <p>This is a simple binomial problem.</p>
<p>Note: I am assuming that both you and the machine choose with replacement. That is, either (or both) of you might have duplicates. </p>
<p>For a single choice you make, the probability that the machine also makes it (as one of the <span class="math-container">$5$</span>... |
1,652,381 | <p>Suppose $f$ derivable on $\mathbb R$ and that $\lim_{x\to a}f'(x)=\ell$. Show that $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\ell.$$</p>
<p>The proof of my course goes like this:</p>
<p>By mean value theorem, there is $y_h\in ]x,x+h[$ s.t. $$f(x+h)-f(x)=f'(y_h)h$$ and thus $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\t... | Enrico M. | 266,764 | <p>To me, it's $40$.</p>
<p>Indeed</p>
<p>$$16 = 2\cdot 8$$
$$27 = 3\cdot 9$$</p>
<p>thence</p>
<p>$$40 = 4\cdot 10$$</p>
<p>The even terms of the series may follow this path, so a possible series could be</p>
<p>$$7, 16, 8, 27, 9, 40, 10, 55, 11, 72, 12, \cdots$$</p>
|
1,652,381 | <p>Suppose $f$ derivable on $\mathbb R$ and that $\lim_{x\to a}f'(x)=\ell$. Show that $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\ell.$$</p>
<p>The proof of my course goes like this:</p>
<p>By mean value theorem, there is $y_h\in ]x,x+h[$ s.t. $$f(x+h)-f(x)=f'(y_h)h$$ and thus $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\t... | Logophobic | 299,855 | <p>Another view, also giving $40$ as the next term: $2n+2;\frac{n}{2};3n+3;\frac{n}{3};4n+4;\frac{n}{4}\cdots$
\begin{align}\text{initial value}&=7\\
7*2+2&=16\\
\frac{16}{2}&=8\\
8*3+3&=27\\
\frac{27}{3}&=9\\
9*4+4&=40\\
\frac{40}{4}&=10
\end{align}
Or: $2(n+1);\frac{n}{2};3(n+1);\frac{n}{3... |
3,378,655 | <p>I have an assignment about epsilon-delta proofs and I'm having trouble with this one. I have worked it through using some of the methods I've picked up for similar proofs but it's just something about this particular expression that doesn't sit right with me. Any feedback would be very helpful. This is how far I've ... | principal-ideal-domain | 131,887 | <p><span class="math-container">$\delta<\min(1,\varepsilon)/2$</span> should do the job.</p>
|
201,994 | <p>We arbitrarily choose n lattice points in a 3-dimensional Euclidean space such that no three points are on the same line. What is the least n in order to guarantee that there must be three points x, y, and z among the n points such that the center of mass of the x-y-z triangle is a lattice point?</p>
| binn | 39,264 | <p>19 according to "A lattice point problem and additive number theory" and its references</p>
|
3,187,628 | <p>On a keyboard 1-9 and A-D , how many possible way to have a 5 digit code with 3 different numbers and two identical letters (order isn't a problem)?</p>
<p>My try : 9C3 x 4C1 = 84 x 4 = 336</p>
| Mohammad Riazi-Kermani | 514,496 | <p>If you write down a few terms you will see that your series is indeed the Taylor series of <span class="math-container">$3\sin x$</span> evaluated at <span class="math-container">$x=\pi/3$</span></p>
<p>Thus the answer is <span class="math-container">$3\sin(\pi/3)$</span></p>
|
1,246,540 | <p>$$ x ^{x } = e^{\ln x^x } $$
$$ \lim_{x\rightarrow 0} x^x = \;? $$</p>
<p>I need to find the limit of x to the power of x as x approaches to 0 using l'Hopital's rule. From previous part there is a hint that I should use the first equation somehow, however I am confused how to rearrange the equation into a fraction ... | Community | -1 | <p>For this limit note that we can write </p>
<p>\begin{equation*}
\lim_{x\to 0} x^x=\lim_{x\to 0}\exp{(\log(x^x))}=\lim_{x\to 0}\exp{x\log(x)}.
\end{equation*}</p>
<p>Factor the exponential function out of the limit due to its continuity, ie</p>
<p>\begin{equation*}
\exp (\lim_{x\to 0}x\log(x))=\exp(\lim_{x\to 0}\f... |
164,987 | <p>Find $(x, a, b, c)$ if $$x! = a! + b! + c!$$</p>
<p>I want to know if there are more solutions to this apart from $(x, a, b, c) = (3, 2, 2, 2)$.</p>
| Old John | 32,441 | <p>If $x\ge 4$ then $a \le x-1$, and we have $a!\le x!/4$, and similarly for $b$ and $c$. Then $a!+b!+c!\le \frac{3}{4}x!$, so there are no solutions with $x\ge 4$, and the other cases can be eliminated quickly.</p>
|
333,265 | <p>Can $|-x^2 | < 1 $ imply that $-1<x<1$?
My steps are as follows?
$$| -x^2| < 1 $$
$$-1<(-x^2)< 1 $$
$$-1<(-x^2)< 1 $$
$$-1<x^2< 1 $$
$$\sqrt{-1}<x< \sqrt 1 $$</p>
<p>I'm actually looking for the radius of convergence for the power series of $\frac{1}{1-x^2}$:</p>
<p>$$\fra... | Ayman Hourieh | 4,583 | <p>The conclusion is correct. However, the steps you took to reach it aren't. The $\sqrt{-1}$ at the end should ring an alarm bell.</p>
<p>A better approach would be to notice that $f(x) = x^2$ is a strictly increasing function on $[0, \infty)$ and strictly decreasing on $(-\infty, 0]$. We know that $f(-1) = f(1) = 1$... |
1,318,884 | <blockquote>
<p>Let S be a piecewise smooth oriented surface in <span class="math-container">$\mathbb{R}^3$</span> with positive oriented piecewise smooth boundary curve <span class="math-container">$\Gamma:=\partial S$</span> and <span class="math-container">$\Gamma : X=\gamma(t), t\in [a,b]$</span> a rectifiable para... | Community | -1 | <p>Here is one way without using the Fundmental theorem of arithmetic just using the definitions </p>
<p>The definition of lcm(a,b) is as follows:</p>
<p>t is the lowest common multiple of a and b if it satisfies the following:</p>
<p>i)a | t and b | t </p>
<p>ii)If a | c and b | c, then t | c.</p>
<p>Similiarly f... |
724,823 | <p>I would like to know if there is some way to imagine the case when a 3D subspace intersects with a 2D plane in a 4D space.
For example, let's have a 3D space in 4D
$$A = \left(\begin{array}{c}1 & 2 & 3 & 4\\7 & 9 & 0 & 5\\ 3 & 3 & 4 & 1\end{array}\right)$$
and
$$B=\left(\begin{arr... | Christoph | 86,801 | <p>Let $e_1, e_2, e_3, e_4$ be the standard basis vectors of $\mathbb R^4$. The $3$-dimensional subspace
$$
U = \operatorname{span}(e_1, e_2, e_4) = \left\{ (x,y,0,w) \in \mathbb R^4 \right\}
$$
and the $2$-dimensional subspace
$$
V = \operatorname{span}(e_3, e_4) = \left\{ (0,0,z,w) \in \mathbb R^4 \right\}
$$
interse... |
569,814 | <p>I was watching a video on YouTube on Quantum Mechanics Concepts and saw that if you wanted to convert a probability amplitude to a probability, you square it. In the video he said that this was equivalent to multiply by it's complex conjugate. So is this correct? Is squaring the same as multiplying by the complex co... | dfeuer | 17,596 | <p>No, they're different, in both math and quantum mechanics. What is true, however, is that $z\bar z=|z|^2$. In QM, the probability is the square of the <em>absolute value</em> of the amplitude.</p>
|
569,814 | <p>I was watching a video on YouTube on Quantum Mechanics Concepts and saw that if you wanted to convert a probability amplitude to a probability, you square it. In the video he said that this was equivalent to multiply by it's complex conjugate. So is this correct? Is squaring the same as multiplying by the complex co... | cderwin | 50,816 | <p>Not exactly. You don't square the complex number itself, you square its <em>magnitude</em>. You can see this is true by actually doing the computation for each value:
Let $z=a+bi$. First the modulus:
$$\left|z\right|^2=\sqrt{a^2+b^2}^2=a^2+b^2$$
Then the conjugate:
$$z\bar{z}=(a+bi)(a-bi)=a^2-abi+abi-b^2i^2=a^2-(... |
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| Will Jagy | 10,400 | <p>Intro: this is the business about taking the gcd of a polynomial and its derivative, in order to detect a repeated factor. In this case, the repeated factor just squares to give the original, so this is one of the simplest types. For this answer yesterday, <a href="https://math.stackexchange.com/questions/1899346/ho... |
2,542,242 | <p>When I tried solving for the inverse of $f(x)=\frac{1-x}{-x}$, I got this:</p>
<p>$f^{-1}(x)=\frac{1}{1-x}$</p>
<p>I know that the way to check my answer would be to take the inverse of the inverse I just found, but this is what I get:</p>
<p>$f^{-1}(f^{-1}(x))=1-\frac{1}{x}=\frac{x-1}{-x}$</p>
<p>The last part,... | Jacky Chong | 369,395 | <p>Another way is by a change of variable, which is also essentially how one prove the Leibniz formula mentioned by @spaceisdarkgreen . Take $u = t/x$, then we see that
\begin{align}
\int^x_0 f(x-t)\ dt = \int^1_0 xf(x(1- u))\ du.
\end{align}
Taking the derivative with respect $x$ gives us
\begin{align}
\frac{d}{dx}\in... |
1,547,057 | <p>I've been working on the following problems, and I know how to integrate functions,but I do not know how to find the value of "c" in the examples below when finding the antiderivative. Any idea what to do? Cheers</p>
<ol>
<li><p>A particle travels in a straight line such that its acceleration at time <em>t</em> sec... | SchrodingersCat | 278,967 | <p><strong>HINT</strong>: Your first problem requires you to find both displacement and velocity. So you have to solve the first order as well as the second order differential equation. For the second order differential equation, you will have 2 constants, say $A$ and $B$. Use the 2 conditions for displacement and solv... |
1,547,057 | <p>I've been working on the following problems, and I know how to integrate functions,but I do not know how to find the value of "c" in the examples below when finding the antiderivative. Any idea what to do? Cheers</p>
<ol>
<li><p>A particle travels in a straight line such that its acceleration at time <em>t</em> sec... | Nizar | 227,505 | <p>For the first problem: you have the acccelaration is $$a(t)=6t+1 $$ Let $x(t)$ and $v(t)$ denote the displacement and the velocity respectively. Then $x''(t)=a(t)$. So integration $a(t) $ twice we end up with $$ x(t)= t^3 +\frac{t^2}{2}+At+B$$ but we have $x(2)=12$ and $x(3)=34$ Then $$ 8+ 2 +2A+B= 12$... |
958,267 | <p>When I was doing problems in my textbook, I came across this problem:</p>
<blockquote>
<p>The velocity of a heavy meteorite entering the earth's atmosphere is inversely proportional to $\sqrt s$ when it is s kilometers from the earth's center. Show that the meteorite's acceleration is inversely proportional to $... | Varun Iyer | 118,690 | <p>The problem is here that we don't know the <strong>time.</strong></p>
<p>However, we do know what $s$ is, since it is in our equation.</p>
<p>Therefore,</p>
<p>we split up the derivative.</p>
<p>Taking the derivative with respect with $t$ for the equation.</p>
<p>$$\frac{dv}{dt}$$</p>
<p>Now, since we know $s$... |
4,338,558 | <p>I am not sure if my proof here is sound, please could I have some opinions on it? If you disagree, I would appreciate some advice on how to fix my proof. Thanks</p>
<p><span class="math-container">$X_1, X_2, ..., X_n$</span> are countably infinite sets.</p>
<p>Let <span class="math-container">$X_1 = \{{x_1}_1, {x_1}... | H A Helfgott | 169,068 | <p>Let me now work out the harder cases, starting as @Ninad_Munshi has suggested.</p>
<p>Let <span class="math-container">$$U(R) = \{(x,y)\in [1,\infty)\times [1,\infty): x^\alpha y^\alpha \sqrt{|x\pm y|}> R\},$$</span>
where <span class="math-container">$\alpha>1$</span>. We want to estimate
<span class="math-c... |
4,003,019 | <p>So I've just started learning topology and a lot of the definitions confuse me.
My main problem is that some of the definitions are quite inconsistent to me for example in topology without tears the author says that a topology <span class="math-container">$\tau$</span> is a set of subsets of <span class="math-contai... | Paul Frost | 349,785 | <p>I think the word "family" is somewhat ambiguous. For example, <a href="https://en.wikipedia.org/wiki/Family_of_sets" rel="nofollow noreferrer">Wikipedia</a> says</p>
<blockquote>
<p>In set theory and related branches of mathematics, a collection <span class="math-container">$F$</span> of subsets of a given... |
2,006,437 | <p>Is it possible to get an example of two matrices $A,B\in M_4(\mathbb{R})$ both having $rank<2$ but $\det(A-\lambda B)\ne 0$ i.e it is not identically a zero polynomial. where $\lambda$ is indeterminate, I mean a variable. I want to say $(A-\lambda B)$ is of full rank matrix, assuming entries of $A-\lambda B$ is a... | RGS | 329,832 | <p>Note that $rank\ A, B < 2$ means their ranks are 1 (If any of them is 0 your question has a trivial answer). That means that all columns of $A$ are $v, \alpha_v v, \beta_v v, \delta_v v$ and the columns of $B$ are $u, \alpha_u u, \beta_u u, \delta_u u$. Now note that the columns of $(A - \lambda B)$ will be sums ... |
215,125 | <p>Suppose $A(x)$ is a $\Delta_0$ formula defining a non-empty set of natural numbers. It's an easy theorem that there is a primitive recursive function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $Range(f) = \{n \in \mathbb{N} \mid A(n)\}$. I'm wondering if it's known whether either of the following strengthenings... | CooLee | 33,232 | <p>I am also interested in the dependency of the constant in the Harnack inequality.
<a href="https://mathoverflow.net/questions/261113/dependency-of-the-constant-in-the-harnack-inequality">Dependency of the constant in the Harnack inequality</a></p>
|
1,552,775 | <p>In how many ways 5 blue pens and 6 black pens can be distributed to 6 children?</p>
<hr>
<p>To do that I used:
$\text{Coefficient of } x^6 \text{ in } ((1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4+x^5+x^6))$
and got answer = 6</p>
<p>but options given are:</p>
<p>a) 97020</p>
<p>b) 116424</p>
<p>c) 8008</p>
<p>d) 67... | Siddharth Joshi | 288,487 | <p>Instead find the coefficient of $x^6 y^5$ in $(1+x+x^2+x^3+...x^\infty)^6(1+y+y^2+y^3+....y^\infty)^5$</p>
<p>I'll explain to you why taking the coefficient gives the answer for distribution problems.
Consider the expression $(1+x+x^2+x^3+...x^\infty)^n$. You can write this expression as follows:$$(1+x+x^2+x^3+...... |
222,312 | <p>How to prove that if you have $x^*$ such that $x^*=\text{psuedoinverse}(A) b$, and $Ay=b$, then
$$\Vert x^* \Vert_2 \leq \Vert y \Vert_2$$</p>
| Community | -1 | <p>You essentially want to find the solution to he following optimization problem.
$$\min_{x}\Vert x \Vert_2 \text{ such that } Ax = b$$
Using Lagrange multipliers, we get that
$$\min_{x, \lambda} \dfrac{x^Tx}2 + \lambda^T (Ax - b)$$
Differentiate with respect to $x$ and $\lambda$ to get that
$$x^* = \underbrace{A^T(A... |
4,482,637 | <p>Is there are a term for a generalized exponential function? The series expansions of sine and cosine look very similar to the exponential function's series expansion</p>
<p><span class="math-container">\begin{align}
& \sum_{n = 0}^{\infty} \frac{x^n}{n!} \\
& \sum_{n = 0}^{\infty} \frac{(-x)^n}{n!} \\
& ... | Empy2 | 81,790 | <p>An integral might run from <span class="math-container">$x=0$</span> to <span class="math-container">$x=X$</span>.</p>
<p>In <span class="math-container">$\int2xdx$</span>, the function is increasing as you go along. So the first <span class="math-container">$2$</span> is there all the way as x goes from 1 to X, bu... |
285,975 | <p>I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time.</p>
<blockquote>
<p>At a distance of <span class="math-container">$45m$</span> from a traffic light, a car traveling <span class="math-container">$15 m/sec$</... | André Nicolas | 6,312 | <p>For (i), prove first that there is an $N$ such that $f(x)$ is <strong>increasing</strong> in the interval $(N,\infty)$, and greater than $1$. Now suppose $a\gt N$ and $f(a)=m$. Then $f(a+m)\gt m$, and $f(a+m)$ is divisible by $m$. </p>
<p>Note that we used the fact (ii) that you were asked to prove. For the proof... |
1,865,086 | <p>$$T_n=2^{-n}$$</p>
<p>How can I tell if this converges? with previous questions I have just let $n = \infty$, however I'm unsure about this one.</p>
| b00n heT | 119,285 | <p>Hint: a monotone decreasing sequence bounded from below converges.</p>
<p>(Similarly a monotone increasing sequence boundes from above converges also)</p>
<p>Of course this is a non constructive approach, as you only get the existance of a limit, and not what the limit actually is. But in some cases, this is alrea... |
1,778,037 | <p>$ 9^x-6^x=4^{x+1/2}$, solve for $x$</p>
<p>Please don't solve this problems entirely, I just want some hints. I have tried to substitute $3^x=b$ and $2^x=a$. </p>
| Shourya Pandey | 331,004 | <p>Try dividing $9^x $ throughout, and substitute $(\frac {2}{3} )^x $ as $y $. Also, notice that $\frac {4}{9} = (\frac {2}{3})^2 $.</p>
|
959,410 | <p>I need to evaluate an expression similar to the following:</p>
<p>$\frac{\partial\mathrm{log}(a+b)}{\partial a}$</p>
<p>At this point I don't know how to proceed. $b$ is a constant so there should be some way to eliminate it.
How would you proceed in this case?</p>
<p>Actually, the original expression is much mor... | Joseph | 158,173 | <p>Clearly, the answer is
$\frac{\partial log(a+b)}{\partial a} = \frac{1}{a+b}$
As you said, b is a constant but in this case it is also in an argument of a derivated function and thus cannot be eliminated using derivatives.</p>
|
2,755,091 | <p>I know that I have to prove this with the induction formula. If proved the first condition i.e. $n=6$ which is divisible by $6$. But I got stuck on how to proceed with the second condition i.e $k$.</p>
| Peter Szilas | 408,605 | <p>Induction step:n+1.</p>
<p>$7^{n+1} -1 = 7\cdot 7^{n} -1=$</p>
<p>$ (6+1)(7^n) -1=$</p>
<p>$6 \cdot 7^n +(7^n -1).$</p>
<p>By hypothesis $(7^n-1)$ is divisible by $6$, hence the above sum is divisible by $6.$</p>
|
3,621,836 | <p>Let's say I have a large determinant with scalar elements:</p>
<p><span class="math-container">$$\begin{vmatrix} x\cdot a & x\cdot b & x\cdot c & x\cdot d \\ x\cdot e & x\cdot f & x\cdot g & x\cdot h \\ x\cdot i & x\cdot g & x\cdot k & x\cdot l \\ x\cdot m & x\cdot n & x\... | Rezha Adrian Tanuharja | 751,970 | <p>Since <span class="math-container">$\omega,\omega^{2}$</span> are the roots of <span class="math-container">$f(x)=x^{2}+x+1$</span>,</p>
<p><span class="math-container">$$
(2-\omega)(2-\omega^{2})(2-\omega^{19})(2-\omega^{23}) =\left(f(2)\right)^{2}=49
$$</span></p>
|
3,238,316 | <p>I think there is a glitch in the <a href="https://en.wikipedia.org/w/index.php?title=Jensen%27s_inequality&oldid=887772941#Proof_1_(finite_form)" rel="nofollow noreferrer">proof by induction</a>. The proof is still valid, but they add an unnecessary assumption:</p>
<p>In the induction step, they choose one of t... | Misha Lavrov | 383,078 | <p>You are right that the case <span class="math-container">$\lambda_1 = 0$</span> is fine; the proof should be corrected to assume instead that <span class="math-container">$\lambda_1<1$</span>. (If <span class="math-container">$\lambda_1=1$</span> the inequality is trivial but should be shown differently, since we... |
244,521 | <p>I cannot prove this statement, I tried to prove by using the definition of open sets however i feel that it is necessary prove it in two directions since it's an iff statement.</p>
<p>The question is, </p>
<p>Let $X$ be a metric space with metric $d$, and let $x_n$ be a sequence of points in $X$. Prove that $x_n\r... | Tom Oldfield | 45,760 | <p>I'm not sure why you've done what you've done, or what it proves.</p>
<p>To prove the if part of your theorem, suppose we have a sequence $x_n$. For any $\epsilon > 0$, there exists $N$ such that $x_n\in B(a,\epsilon)$ $\forall n > N $ (since $B(a,\epsilon)$ is an open set containing $a$) . Hence $x_n \righta... |
3,774,263 | <ol>
<li>Is it possible that distance (<span class="math-container">$r$</span>) or angle (<span class="math-container">$θ$</span>) contains Imaginary or Complex number?</li>
<li>If the answer is yes, how can I convert a number like that (Polar with complex argument) to Rectangular number? <br />
For example:
<strong><s... | Aaron | 9,863 | <p>Here are two perspectives that are slightly different from the answers already given.</p>
<p>Since <span class="math-container">$X$</span> is of dimension greater than <span class="math-container">$1$</span>, we can find two linearly independent vectors <span class="math-container">$v,w\in X$</span>. If <span class... |
1,726,745 | <p>I am looking for information in regards to a couple particular functions: </p>
<p>1) $P(x)=\sum_{p\in\mathbb{P}}\frac{x^p}{p!}$</p>
<p>2) $Q(x)=\sum_{p\not\in\mathbb{P}}\frac{x^p}{p!}$ (assuming $0, 1$ are included powers in the series...)</p>
<p>3) $R(x)=\frac{1}{P(x)}$</p>
<p>4) $S(x)=\frac{1}{Q(x)}$</p>
... | Plutoro | 108,709 | <p>From the comment chain on the question, the OP says that some plots would be helpful, so though I cannot answer this question, this is what I can say.</p>
<p>Though finding an analytic expression for these seems a monumental task, approximating them in the first several terms or so is quite easy. I evaluated each o... |
13,861 | <p>I am currently working with a weighted adjacency matrix for a directed graph, and it contains several 0 columns and rows. With the unaltered matrix, I am able to monitor the relations between vertices with,</p>
<pre><code>TableForm[Normal @ WeightedAdjacencyMatrix[graph],
TableHeadings -> {a = VertexList[gr... | Dr. belisarius | 193 | <pre><code>graph= Graph[Range@5 , {1 -> 5, 5 -> 3, 3 -> 1}, EdgeWeight-> RandomInteger[100, 3]]
shAdj[graph_] :=
Grid[Transpose[
Select[Transpose[
Select[Join[{Join[{""}, VertexList[graph]]},
Transpose[Join[{VertexList[graph]}, WeightedAdjacencyMatrix[graph]]]],
Total@Rest@# !... |
77,290 | <p>How would you determine all integers $m$ such that the following is true? </p>
<p>$$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$</p>
<p>Note that $\lfloor \cdot \rfloor$ means the greatest integer function. Also, $x$ must be a positive real number.</p>
| N. S. | 9,176 | <p>Here is an idea, the computations seem too long to actually try.</p>
<p>Naive approach: forget the integer part. We need then $m=\frac{10x}{7}$ or $x = \frac{7m}{10}$.</p>
<p>Now non-naive approach: Try $x = \frac{7m+k}{10}$ where $k$ is small enough. $0 \leq k \leq 9$ or $6$ should work depending by whatever $7m$... |
77,290 | <p>How would you determine all integers $m$ such that the following is true? </p>
<p>$$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$</p>
<p>Note that $\lfloor \cdot \rfloor$ means the greatest integer function. Also, $x$ must be a positive real number.</p>
| Steven Alexis Gregory | 75,410 | <p>Consider the reciprocal equation
$$ m = \dfrac{\lfloor 2x \rfloor \cdot \lfloor 5x \rfloor}
{\lfloor 2x \rfloor + \lfloor 5x \rfloor}$$</p>
<p>Letting $x=n+\delta$ where $0 \le \delta < 1$, we get</p>
<p>$$ m = \dfrac{(2n + \lfloor 2\delta \rfloor)(5n + \lfloor 5\delta \rfloor)}
{7n +... |
254,926 | <p>I was working on a problem involving perturbation methods and it asked me to sketch the graph of $\ln(x) = \epsilon x$ and explain why it must have 2 solutions. Clearly there is a solution near $x=1$ which depends on the value of $\epsilon$, but I fail to see why there must be a solution near $x \rightarrow \infty$... | ncmathsadist | 4,154 | <p>Put $$f(x) = {\ln(x)\over x}.$$<br>
Then you have
$$f'(x) = {1 - \ln(x)\over x^2}.$$
When $x > e$, the function $f$ decreases; in fact is is easy to see it decreases to zero as $x\to\infty$. When $x < e$, $f$ increases. You can see that it has the $y$-axis as a vertical asymptote. </p>
<p>The graph has a g... |
254,926 | <p>I was working on a problem involving perturbation methods and it asked me to sketch the graph of $\ln(x) = \epsilon x$ and explain why it must have 2 solutions. Clearly there is a solution near $x=1$ which depends on the value of $\epsilon$, but I fail to see why there must be a solution near $x \rightarrow \infty$... | M. Strochyk | 40,362 | <p>For all $\varepsilon>0$ using L'Hospital's rule
$$\lim\limits_{x \to +\infty} {\dfrac{\varepsilon x}{\ln{x}}}=\varepsilon \lim\limits_{x \to +\infty} {\dfrac{x}{\ln{x}}}=\varepsilon \lim\limits_{x \to +\infty} {\dfrac{1}{\frac{1}{x}}}=+\infty.$$</p>
|
3,792,983 | <p>As far as I can tell the definition of a source and a sink respectively are given in terms of the divergence operator.</p>
<p>That is, given a vector field <span class="math-container">$\vec{D}$</span>, it has a <em>source</em> in point <span class="math-container">$P$</span> if its divergence <span class="math-cont... | glS | 173,147 | <p>Here's a slight rewording of <a href="https://math.stackexchange.com/a/3793101/173147">the other answer</a>.</p>
<p>I want to prove that a closed, convex, nonempty <span class="math-container">$A\subset\mathbb R^n$</span> which contains no lines, always contains at least an extreme points.</p>
<p>The <span class="ma... |
3,624,732 | <p>I'm studying logical operators for school and there's a weird question that keeps bugging me even though it seems pretty basic.
<br>
I was asked to evaluate the proposition : <strong>p -> q -> r</strong> with p, r are False and q is True.
<br>
I tried evaluating it from left to right like this: <strong>( ( p -> q ) ... | Wuestenfux | 417,848 | <p>Well, the operations <span class="math-container">$\wedge,\vee,\Leftrightarrow$</span> are left-associative while the operation <span class="math-container">$\Rightarrow$</span> is right-associative.
So</p>
<p><span class="math-container">$[p\Rightarrow q\Rightarrow r ]\Longleftrightarrow [p\Rightarrow (q\Rightarro... |
3,001,860 | <p>I am looking for a <span class="math-container">$u(x)$</span> and an <span class="math-container">$f$</span> that make the answer to <span class="math-container">$$u(x)''''-u(x)''+u(x)=f(x)$$</span> where <span class="math-container">$$u'(-1)=u'(1)=u(1)=u(-1)=0$$</span> reasonably short.</p>
<p>I have some numerica... | Aleksas Domarkas | 562,074 | <p>If
<span class="math-container">$$u(x)={{\cos{\left( \frac{\pi x}{2}\right) }}^{2}}$$</span>
then
<span class="math-container">$$f(x)=(\pi^4+\pi^2+1){{\cos{\left( \frac{\pi x}{2}\right) }}^{2}}-\frac{\pi^4}{2}-\frac{\pi^2}{2}.$$</span></p>
|
2,838,537 | <p>Is the function<br>
$$f(x)=\begin{cases} x+0.2x^2 \sin (1/x) & x \ne 0\\ 0 & x=0 \end{cases}$$invertible in a neighborhood of origin?</p>
<p>I just know that $f$ is continuous on $\mathbb{R}$ and
$$f'(x)=\begin{cases}
1+0.4x\sin\left(\frac{1}{x}\right)-0.2\cos\left(\frac{1}{x}\right)& x \ne 0\\
1 &... | Fred | 380,717 | <p>Hint: we have $f'(x)> 0.4>0$ for all $x$.</p>
<p>Conclusion ?</p>
|
2,838,537 | <p>Is the function<br>
$$f(x)=\begin{cases} x+0.2x^2 \sin (1/x) & x \ne 0\\ 0 & x=0 \end{cases}$$invertible in a neighborhood of origin?</p>
<p>I just know that $f$ is continuous on $\mathbb{R}$ and
$$f'(x)=\begin{cases}
1+0.4x\sin\left(\frac{1}{x}\right)-0.2\cos\left(\frac{1}{x}\right)& x \ne 0\\
1 &... | Crostul | 160,300 | <p>Your computations of $f'$ tell you that in the set $[-1;0) \cup (0;1]$ you have
$$f'(x) \ge 1-0.2-0.4=0.4$$</p>
<p>Note that
$f(x)=x(1+x \sin (1/x))$
is equal to $x$ times a strictly positive quantity. Thus </p>
<ol>
<li>$f(x) <0$ if and only if $x<0$;</li>
<li>$f(x) >0$ if and only if $x>0$;</li>
<li... |
4,065,655 | <p>If <span class="math-container">$A B = C$</span> where matrix <span class="math-container">$C$</span> is stochastic (all entries are positive and all rows add to <span class="math-container">$1$</span>) then is it necessary that both <span class="math-container">$A$</span> and <span class="math-container">$B$</span>... | Peter Melech | 264,821 | <p>First note that
<span class="math-container">$$\int_0^1x^{t-1}dx$$</span>
converges and then use that <span class="math-container">$|x^{t-1}e^{-x}|\leq x^{t-1}$</span> for <span class="math-container">$x\in[0,1]$</span>.
Then see that
<span class="math-container">$$\int_1^{\infty}x^{-2}dx$$</span>
converges and use ... |
1,597,332 | <p>Get the following quadratic form:</p>
<p>$$Q(x)=x_1^{2}+x_3^2+4x_1x_2-4x_1x_3 $$</p>
<p>to obtain the canonical form I tried the following: </p>
<p>$$Q(x)=x_1^{2}+x_3^{2}+4x_1x_2-4x_1x_3=4(x_1^{2})+(x_3^{2})-4x_1x_3-3x_1x_1+4x_1x_2-(4/3)x_2x_2+(4/3)x_2x_2=(2x_1-x_3)^2-... $$</p>
<p>There I stopped because I reme... | Rene Schipperus | 149,912 | <p>I am not at all sure this is what you are looking for but dont you complete the square:</p>
<p>$$x_1^{2}+x_3^2+4x_1x_2-4x_1x_3=x_1^{2}+(4x_2-4x_3)x_1 +x_3^2$$
$$=x_1^{2}+(4x_2-4x_3)x_1 +(2x_2-2x_3)^2 -(2x_2-2x_3)^2 + x_3^2$$</p>
<p>$$=(x_1+2x_2-2x_3)^2 -(2x_2-2x_3)^2 + x_3^2$$</p>
<p>$$=(x_1+2x_2-2x_3)^2 -4(x_2^2... |
1,597,332 | <p>Get the following quadratic form:</p>
<p>$$Q(x)=x_1^{2}+x_3^2+4x_1x_2-4x_1x_3 $$</p>
<p>to obtain the canonical form I tried the following: </p>
<p>$$Q(x)=x_1^{2}+x_3^{2}+4x_1x_2-4x_1x_3=4(x_1^{2})+(x_3^{2})-4x_1x_3-3x_1x_1+4x_1x_2-(4/3)x_2x_2+(4/3)x_2x_2=(2x_1-x_3)^2-... $$</p>
<p>There I stopped because I reme... | Will Jagy | 10,400 | <p>There is a method I asked about at
<a href="https://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr">reference for linear algebra books that teach reverse Hermite method for symmetric matrices</a><br>
The main advantage is that it is a recipe... |
1,811,567 | <p>We had a high school mathematics teacher who taught us a cool technique that I've forgotten. It can be used, for example, for developing a formula for the sum of squares for the first "n" integers. You start by making a column for Sn, and then determine the differences until you get a constant. See the picture.</p>
... | Jack D'Aurizio | 44,121 | <p>Every polynomial that takes integer values over the integers can be represented with respect to the binomial base as a linear combination with integer coefficients. In our case:</p>
<p>$$ n^2 = \color{blue}{2}\binom{n}{2}+\color{blue}{1}\binom{n}{1}+\color{blue}{0}\binom{n}{0} \tag{1}$$
And that leads to:
$$ \sum_{... |
28,348 | <p>Thomson et al. provide a proof that <span class="math-container">$\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$</span> in <a href="http://classicalrealanalysis.info/documents/TBB-AllChapters-Landscape.pdf#page=95" rel="noreferrer">this book (page 73)</a>. It has to do with using an inequality that relies on the binomial... | JavaMan | 6,491 | <p>Since $x \mapsto \log x$ is a continuous function, and since continuous functions respect limits:
$$
\lim_{n \to \infty} f(g(n)) = f\left( \lim_{n \to \infty} g(n) \right),
$$
for continuous functions $f$, (given that $\displaystyle\lim_{n \to \infty} g(n)$ exists), your proof is entirely correct. Specifically,
... |
28,348 | <p>Thomson et al. provide a proof that <span class="math-container">$\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$</span> in <a href="http://classicalrealanalysis.info/documents/TBB-AllChapters-Landscape.pdf#page=95" rel="noreferrer">this book (page 73)</a>. It has to do with using an inequality that relies on the binomial... | marty cohen | 13,079 | <p>Here's a two-line, completely elementary proof that uses only Bernoulli's inequality:</p>
<p>$$(1+n^{-1/2})^n \ge 1+n^{1/2} > n^{1/2}$$
so, raising to the $2/n$ power,
$$ n^{1/n} < (1+n^{-1/2})^2 = 1 + 2 n^{-1/2} + 1/n < 1 + 3 n^{-1/2}.$$</p>
<p>I discovered this independently, and then found a very simi... |
2,010,423 | <p>Are $$(a, b, c) = (\mid 1 \mid, \mid 2 \mid, \mid 2 \mid), (\mid 2 \mid, \mid 4 \mid, \mid 4 \mid), (\mid 2 \mid, \mid 3\mid, \mid 6 \mid), (\mid 1 \mid, \mid 1 \mid, \mid 1 \mid)$$ the only integers such that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} $$ is an integer ?</p>
| DLIN | 355,583 | <p>We can consider the positive case, i.e. $a,b,c>0$.</p>
<p>We can categorize into the 3 cases(from simple to copmlex):</p>
<p>1): $a=b=c$. Then $\frac1a+\frac1b+\frac1c=\frac3a\in\mathbb Z$, so $a|3$, hence $a=1,3$. </p>
<p>2): $0<a<b<c$. Then $1\leq\frac1a+\frac1b+\frac1c<\frac3a$, $1\leq a<3$, ... |
2,240,616 | <p>Let $f:[0,1] \to \mathbb{R}$ be a function such that for every $a \in [0,1)$ and $b \in (0,1]$ the one-sided limits $$f(a^+)=\lim _{x\to a^+}f(x) \in \mathbb{R}$$ $$f(b^-)=\lim _{x \to b^-} f(x) \in \mathbb {R}$$ exist. </p>
<p>A) Show that $f$ is bounded. </p>
<p>B) Does $f$ necessarily achieve its maximum at som... | DanielWainfleet | 254,665 | <p>For (C).</p>
<p>Suppose $f(0)>0$ (as the case $f(0)<0$ is handled similarly). Since $f$ is continuous at $0$, there exists $r\in (0,1)$ such that $\forall x\in [0,r)\;(f(x)>f(0)/2).$ So there exists $r\in (0,1)$ such that $\forall x\in [0,r)\; (f(x^-)\geq 0\land f(x+)\geq 0).$ </p>
<p>Let $P(r)\iff \fora... |
2,663,370 | <p>Consider the following integral:</p>
<p>$$
I(k):= \int_k^{+\infty} \frac{e^{-1/x} \log(x)}{x^2} dx\\
=-\int_0^{1/k} e^{-t} \log t \, dt,
$$</p>
<p>where $k>0$. It is known that $\lim_{k \to 0} I(k)=\gamma$, where $\gamma$ denotes the Euler-Mascheroni constant. Henceforth, it must be that $\lim_{k\to +\infty}I(k... | Kelenner | 159,886 | <p>Hint: For $x>0$, put $F(x)=\int_0^x\exp(-t)\log tdt$, integrate by parts using that $1-\exp(-t)$ is a primitive of $\exp(-t)$, and show that
$$F(x)=(x\log x)(\frac{1-\exp(-x)}{x}-\frac{1}{x\log x}\int_0^x \frac{1-\exp(-t)}{t}dt)$$</p>
|
718,901 | <p>I have for a very long time been thinking about certain notations in differential calculus, for instance it might sometimes be sufficient to carry out a variable substitution in integrals. For example we put $t=x^2$ and then write $\frac {dt}{dx} = 2x \Leftrightarrow dt = (2x)dx$</p>
<p>I dont quite understand the ... | Henry | 6,460 | <p>To take an example, suppose you wanted to find $$\int_{x=0}^{\frac12} x(1-x^2)^3 dx.$$ How would you interpret $dx$ in that expression?</p>
<p>Would you be willing to do the substitution $t=x^2$ with $\frac{dt}{dx}=2x$ so that you got something like this?</p>
<p>$$\int_{x=0}^{\frac12} x(1-x^2)^3 dx = \int_{x=0}^{... |
54,088 | <p>I am studying KS (Kolmogorov-Sinai) entropy of order <em>q</em> and it can be defined as</p>
<p>$$
h_q = \sup_P \left(\lim_{m\to\infty}\left(\frac 1 m H_q(m,ε)\right)\right)
$$</p>
<p>Why is it defined as supremum over all possible partitions <em>P</em> and not maximum? </p>
<p>When do people use supremum and w... | Brooke Knight | 13,777 | <p>Supremum need not be attained while maximum is. When maximum exist, maximum=supremum.</p>
|
2,497,072 | <p>Write using logical symbols: "Every even number greater than two is the sum of two prime numbers" <br>
This is how I attempted to write this:
$$((n=2m \land n > 2) \Rightarrow n=p_1+p_2)\land(x|p_1 \Rightarrow x=1 \lor x=p_1) \land (y|p_2 \Rightarrow y=1 \lor y=p_2) \land (p_1 >1 \land p_2 >1)$$
I am not su... | Redsbefall | 97,835 | <p>Slightly different view : $$ P := \{ p \: \mid p \in \mathbb{N} \rightarrow \ \forall q \in \mathbb {N } : \: 1<q<p , \: mod(p,q) \ne 0 \: \}$$</p>
<p>$$ n = 2(m+1), \: m \in \mathbb{N} \implies \exists p_{1}, p_{2} \in P \rightarrow n =p_{1}+p_{2}$$</p>
|
2,417,029 | <p>I have been told that a line segment is a set of points.
How can even infinitely many point, each of length zero, can make a line of positive length?</p>
<p>Edit:
As an undergraduate I assumed it was due to having uncountably many points.
But the Cantor set has uncountably many elements and it has measure $0$.</p>
... | Mikhail Katz | 72,694 | <p>You write: "As an undergraduate I assumed it was due to having uncountably many points. When I learned about measure theory, I realized that's not the explanation."</p>
<p>But in fact this <em>is</em> indeed the explanation. Lebesgue measure $\lambda$ is countably additive (in ZFC) so if $\mathbb R$ were countable... |
545,771 | <blockquote>
<p>let sequence $\{a_{n}\}$,such $a_{1}=0,a_{2}=2,a_{3}=5$,and for $n\in N^{+}$,such
$$\begin{cases}
a_{2n}=2n+2a_{n}\\
a_{2n+1}=2n+1+a_{n}+a_{n+1}
\end{cases}$$
How can I find the closed form of $\{a_{n}\}$?</p>
</blockquote>
<p>My try:
we have</p>
<blockquote>
<p>$$a_{2n+1}-a_{2n}=a_{n+1}-a_{n}... | Obinna Okechukwu | 11,283 | <p>Here is a closed form formula for $a_n$ <br>
$$a_n = (\lfloor \log_2{n}\rfloor + 2)n - 2^{\lfloor \log_2{n}\rfloor +1}$$
To get this simply notice that $$a_n = n + a_{\lfloor \frac{n}{2}\rfloor}+ a_{\lceil\frac{n}{2}\rceil}$$
Then follow Greg Martin's sugestions.</p>
|
1,775,378 | <p>So this question is inspired by the following thread: <a href="https://forums.factorio.com/viewtopic.php?f=5&t=25008">https://forums.factorio.com/viewtopic.php?f=5&t=25008</a></p>
<p>In it, the poster is examining an $8$-belt balancer (more on that to come) which he shows fails to satisfy a desirable proper... | Bomaz | 354,628 | <p>This is not a complete answer but</p>
<p>Assumption 1: a unlimited <a href="http://visionfear.com/Pictures/upload/2016/01/10/20160110151931-a269f11e.png" rel="nofollow noreferrer">4:4 balancer is possible</a>. My belief is that this is throughput unlimited.</p>
<p>Take the design of the 4:4 balancer, replace each 2... |
2,843,337 | <blockquote>
<p>If $0 \lt x \lt \dfrac{\pi}{2}$, prove that
$$x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$$</p>
</blockquote>
<p>This question must be done without calculus. First, I tried splitting it into the intervals $(0,\pi/4)$ and $(\pi/4, \pi/2)$, hoping that, $\sin x$ was bound tightly enough on the interva... | Mohammad Riazi-Kermani | 514,496 | <p>Note that $$(9-x^3)/9 +x^3/9 =1$$</p>
<p>Let $$(9-x^3)/9 =\sin ^2 t$$ and $$x^3/9 =\cos ^2 t$$</p>
<p>Thus $$ \cos t \sin x +\sin t \cos x =\sin (x+t) \le 1$$</p>
<p>$$ x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$$</p>
|
394,489 | <p>$$\int^L_{-L} x \sin(\frac{\pi nx}{L})$$</p>
<p>I've seen something like this in Fourier theory, but I'm still not sure how to approach this integral. Wolfram Alpha gives me the answer, but no method. Integrate by parts? Substitution?</p>
| Mark McClure | 21,361 | <p>For what it's worth, WolframAlpha <em>will</em> show you the steps to evaluate the <em>indefinite</em> integral associated with this definite integral. It's then just a matter of plugging in the limits and subtracting.</p>
<p>Here are the steps returned by WolframAlpha via the Mathematica interface:</p>
<p><img s... |
162,364 | <p>Assuming we have a list below, it has real number elements and complex numbers, how can I quickly find if the list has any Real number that its value is less than 1.0? </p>
<pre><code> lis = {3 Sqrt[354], Sqrt[2962], Sqrt[2746], 3 Sqrt[282], Sqrt[2338],
Sqrt[2146], 3 Sqrt[218], Sqrt[1786], Sqrt[1618], 27 Sqrt[2]... | Edmund | 19,542 | <p>You may use <a href="http://reference.wolfram.com/language/ref/Select.html" rel="nofollow noreferrer"><code>Select</code></a>and <a href="http://reference.wolfram.com/language/ref/Element.html" rel="nofollow noreferrer"><code>Element</code></a>.</p>
<pre><code>Select[# ∈ Reals && # < 1 &]@lis
</code>... |
2,426,394 | <p>I'm sorry to bother you with this easy problem. But I'm working alone and totally confused.
It is the 1378th Problem from "Problems in Mathematical Analysis" written by Demidovich. The standard answer is
$$\frac{60(1+x)^{99}(1+6x)}{(1-2x)^{41}(1+2x)^{61}}$$</p>
<p>But when I followed the rutine
$$\frac{d(P(x)/Q(x))... | user577215664 | 475,762 | <p>Or write it this way too it's easier </p>
<p>$P(x)=\frac{(1+x)^{100}}{(1-2x)^{40}(1+2x)^{60}}$</p>
<p>$P(x)=(\frac{(1+x)^{10}}{(1-4x^2)^{4}(1+2x)^{2}})^{10}$</p>
<p>$P(x)=(\frac{(1+x)^{5}}{(1-4x^2)^{2}(1+2x)})^{20}=Q(x)^{20}$</p>
<p>Then apply $Q'(x)^n=nQ(x)^{n-1}Q'(x)$</p>
|
4,292,822 | <blockquote>
<p>Prove linear independence of <span class="math-container">$1+x^3-x^5,1-x^3,1+x^5$</span> in the Vector Space of Polynomials</p>
</blockquote>
<p>The attempts I found online all are quite easy. You just substitute something in for <span class="math-container">$x$</span> into the equation <span class="mat... | Servaes | 30,382 | <p>The reason that you can substitute and differentiate, is because every polynomial defines a function from <span class="math-container">$\Bbb{R}$</span> to <span class="math-container">$\Bbb{R}$</span>. The set <span class="math-container">$\operatorname{Map}(\Bbb{R},\Bbb{R})$</span> of all functions from <span class... |
1,137 | <p>In a freshman course in mathematics for biology students, I have had the issue that basic algebra (e.g. simplifying $\frac{a/b}{c/d}$) is far from being mastered. On the one hand, it seems ill-suited to teach more advanced matters to student failing on the basics. On the other hand, if we take too much time to cover... | user568 | 568 | <p>This is a situation where using an online homework system is particularly useful; usually there will be an early section of review material from which you can assign problems. I would advise issuing such an assignment as early as possible in the term (preferably on the first day), making sure the students realize it... |
3,421,969 | <p>The point <span class="math-container">$C = (1, 2)$</span> lies inside the circle <span class="math-container">$x^2 + y^2 = 9$</span>. What is the length of the shortest chord of the circle through <span class="math-container">$C$</span>?</p>
| Heatconomics | 531,927 | <p>Let <span class="math-container">$S=1+a+a^2+\dots+a^{n-1}$</span>. Then <span class="math-container">$aS=a+a^2+\dots+a^n$</span>. Then, <span class="math-container">$S-aS=(1-a)S=1-a^n$</span>. Thus, if <span class="math-container">$a\neq 1$</span>, <span class="math-container">$S=\frac{1-a^n}{1-a}$</span>.</p>
|
3,421,969 | <p>The point <span class="math-container">$C = (1, 2)$</span> lies inside the circle <span class="math-container">$x^2 + y^2 = 9$</span>. What is the length of the shortest chord of the circle through <span class="math-container">$C$</span>?</p>
| Klaus | 635,596 | <p>You made a mistake in the first displayed line, it should be
<span class="math-container">$$a^{n-1}\left(\frac{1-(\frac{1}{a})^n}{1-\frac{1}{a}}\right) = \frac{a^n}{a}\left(\frac{1-\frac{1}{a^n}}{1-\frac{1}{a}}\right) = \left(\frac{a^n-1}{a-1}\right)$$</span>
as desired.</p>
|
1,867,695 | <blockquote>
<p>$\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$</p>
</blockquote>
<p>$\lim_{n\rightarrow \infty} n *\lim_{n\rightarrow \infty}\left ( 1-\sqrt{1-\frac{5}{n}} \right ) = \infty * \left ( 1-\sqrt{1-0} \right ) = \infty * 0 = 0$</p>
<p>Did I do it correctly?</p>
<p>Problem is when ... | Clement C. | 75,808 | <p>In short: <strong>no</strong>, you did not do it correctly. The reason is that $\infty\cdot 0$ is one of the <strong>indeterminate</strong> forms, you cannot conclude it's equal to zero.</p>
<p>The rationale being: <em>it can be anything.</em> For instance, $$\begin{align}
\underbrace{n}_{\to \infty}\cdot \underbra... |
1,867,695 | <blockquote>
<p>$\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$</p>
</blockquote>
<p>$\lim_{n\rightarrow \infty} n *\lim_{n\rightarrow \infty}\left ( 1-\sqrt{1-\frac{5}{n}} \right ) = \infty * \left ( 1-\sqrt{1-0} \right ) = \infty * 0 = 0$</p>
<p>Did I do it correctly?</p>
<p>Problem is when ... | Hulkster | 212,102 | <p>$$n-n\sqrt{1-\frac{5}{n}}=\frac{(n-n\sqrt{1-\frac{5}{n}})(n+n\sqrt{1-\frac{5}{n}})}{n+n\sqrt{1-\frac{5}{n}}}=\frac{5}{1+\sqrt{1-\frac{5}{n}}}\rightarrow \frac{5}{2}, \space \text{as} \space \space n \rightarrow \infty.$$</p>
|
2,823,675 | <p>Let $f$ nonnegative continuous decreasing function on $[0,\infty)$.
Suppose $\frac{f(x)}{\sqrt{x}}$ is integrable, then $\sqrt{x}f(x) \to 0$ as $x\to \infty$</p>
<p>I tried to prove $\sqrt{x}f(x)$ is Cauchy sequence, but I cannot to do. And I don't know how to use continuity.</p>
| Bill O'Haran | 551,590 | <p>Let's just shift the problem to make it easier:</p>
<blockquote>
<p>Let $f: \mathbb{R}^+ \to \mathbb{R}$ be a continuous, decreasing and integrable (on $\mathbb{R}^+$) function.
Prove that $xf(x)\to 0$ as $x\to \infty$.</p>
</blockquote>
<p><strong>First step:</strong> Notice that $f\to 0$ as $x\to \infty$ (fa... |
2,823,675 | <p>Let $f$ nonnegative continuous decreasing function on $[0,\infty)$.
Suppose $\frac{f(x)}{\sqrt{x}}$ is integrable, then $\sqrt{x}f(x) \to 0$ as $x\to \infty$</p>
<p>I tried to prove $\sqrt{x}f(x)$ is Cauchy sequence, but I cannot to do. And I don't know how to use continuity.</p>
| mathcounterexamples.net | 187,663 | <p>Take $0<u<v$. You have for $x \in [u,v]$ $\frac{f(u)}{\sqrt{x}} \le \frac{f(x)}{\sqrt{x}} \le \frac{f(v)}{\sqrt{x}} $</p>
<p>Hence</p>
<p>$$0 \le 2\left(\sqrt{v}-\sqrt{u}\right) f(u)\le \int_u^v F(t) \ dt\le 2\left(\sqrt{v}-\sqrt{u}\right) f(v)$$
where $F(x) = \frac{f(x)}{\sqrt{x}}$</p>
<p>Now take $u=x$ an... |
481,017 | <blockquote>
<p>Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$.</p>
</blockquote>
<p>Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the text of the problem does not specify further. </p>
<p><strong>Possibly Helpful L... | Michael Albanese | 39,599 | <p>The following observation comes from a comment by Sergei Ivanov in <a href="https://mathoverflow.net/questions/17614/solving-ffx-gx">this</a> MathOverflow post. It discusses the existence of $f : \mathbb{C} \to \mathbb{C}$ such that $f(f(z)) = g(z)$ where $g(z) = z^2 - 2$. I am aware that this is not precisely what ... |
481,017 | <blockquote>
<p>Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$.</p>
</blockquote>
<p>Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the text of the problem does not specify further. </p>
<p><strong>Possibly Helpful L... | Steven Stadnicki | 785 | <p>Any solutions to this problem should be findable via conjugacy; the key is that the given quadratic is <em>topologically conjugate</em> to the 'critical' logistic map, which in turn is known to be topologically conjugate to the so-called <em>bit shift</em> map. Let $f(x) = x^2-2$, $g(x) = 4x(1-x)$, $p(x) = 2-4x$. ... |
481,017 | <blockquote>
<p>Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$.</p>
</blockquote>
<p>Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the text of the problem does not specify further. </p>
<p><strong>Possibly Helpful L... | mercio | 17,445 | <p>Michael's answer is fine, but here is a more in-depth analysis.</p>
<p>We want to look at the connected components of the graph whose vertices are the reals and where there is an arrow from $x$ to $y$ when $y = f^2(x) = x^2-2$.</p>
<p>It is clear that $f$ has to send connected components to connected components, s... |
1,814,474 | <p>Lets have a series
$$\sum_{n=2}^\infty\frac{\ln\left(\frac{n+1}{n-1}\right)}{\sqrt{n}}$$</p>
<p>However, I have absolutely no clue how to try to continue. I could probably use the integral criterion and integrate the problem using the residue theorem, but that is too much of a hassle. Is there an easy way to prove ... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
3,687,355 | <p>I have trouble with how to find eigenvectors when you have a complex eigenvalue</p>
<p>For example the matrix
<span class="math-container">$$ \begin{pmatrix}
0 & 1\\
-2 & -2
\end{pmatrix}$$</span></p>
<p>Here you get the eigenvalues <span class="math-container">$-1$</span> and <span class="math-container">... | Minus One-Twelfth | 643,882 | <p><span class="math-container">$\newcommand{\x}{\mathbf{x}}$</span>You use the same procedure to find the eigenvectors as in the real case, just now you may have to use complex number arithmetic.</p>
<p>Let <span class="math-container">$A = \begin{pmatrix}
0 & 1\\
-2 & -2
\end{pmatrix}.$</span> I'll get you s... |
872,893 | <p>How can I prove that the equation $r=16^r$ is wrong for any arbitrary value of $r$?
I have tried:</p>
<p>\begin{align}
&r=16^r &&\implies \log_r r = \log_r 16^r \\
&&& \implies 1 = r\log_r 16 \\
&&& \implies 1/r = \log_r 16 \\
&&& \implies r^{1/r} = r^{\log_r 16}\\
&a... | Community | -1 | <p>I'm not too good with number theory, so I went about this in a different way and took a calculus-based approach.</p>
<p>Consider the function $f:(0,\infty) \to \mathbb{R}$ defined by $f(x) = x^{1/x}$. Then $$f'(x) = -x^{\frac{1}{x} - 2}(ln(x) - 1).$$</p>
<p>Then $f'(x) > 0$ for $x \in (0, e)$ and $f'(x) < 0... |
872,893 | <p>How can I prove that the equation $r=16^r$ is wrong for any arbitrary value of $r$?
I have tried:</p>
<p>\begin{align}
&r=16^r &&\implies \log_r r = \log_r 16^r \\
&&& \implies 1 = r\log_r 16 \\
&&& \implies 1/r = \log_r 16 \\
&&& \implies r^{1/r} = r^{\log_r 16}\\
&a... | lemon | 89,931 | <p>$16^{r}>0$ for all r and so, for $r=16^{r}$ to hold, you must have $r>0$. </p>
<p>Then observe from the Taylor expansion:
$$ 16^r-r=1+\underbrace{r(\log (16)-1)}_{>0}+\sum_{k=2}^\infty \underbrace{\frac{\log^k(16)}{k!}r^k}_{>0} > 0 $$
for all $r>0$.</p>
|
2,145,412 | <p>If you have a vector line m with equation $r = i + j + 2k + s( 3i + j - k)$
how do you find the normal?</p>
<p>If you presume that normal is $n = ( n i + o j + p k )$ then</p>
<p>$$3n + o -p = 0 $$</p>
<p>but this leaves you with too many options $\left \lbrace (0,0,0), (1,0,3), (0,1,1), ....\right \rbrace$ is i... | Joffan | 206,402 | <p>Computing the factor sum $\sigma_1(N)$ is quickly done from the prime decomposition of $N$. </p>
<p>Suppose $N$ has a number of prime factors $p_1^{a_1}p_2^{a_2}\ldots$. For the purpose of illustration take $a_1=3$. Then for any divisor $d$ of $N$ that is coprime to $p_1$, $N$ is also divisible by $p_1d$, $p_1^2d$,... |
1,314,802 | <p>True.</p>
<p>Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume:</p>
<p>$$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$</p>
<p>Since $ \lim_{n\to\infty} \frac{1}{n} $ is bounded in $ (0,1] $ and $ I \subset (0,1] $, we have by hypothesis $f$ uniformly continuous the... | Bernard | 202,857 | <p>$$\frac{1-\cos x}x=\frac x2+o(x)$$
hence
$$\ln\Bigl(1+\frac{1-\cos x}x\Bigr)^{\!\tfrac 1x}=\frac1x\ln\Bigl(1+\frac x2+o(x)\Bigr)=\frac1x\Bigl(\frac x2+o(x)\Bigr)=\frac12+o(1)$$
so that the limit is $\,\mathrm e^{1/2}$.</p>
|
1,301,763 | <p>The only problem I have with this is knowing when you are solving for dx or dy. For example, this question which says</p>
<p>find the volume of the solid created by rotating the region bounded by y = 2x-4, y = 0, and x = 3, about the line x = 4. This one is dy. Instead of y = 2x-4, the person solved in respect to x... | Peter | 82,961 | <p>If the piece of the function rotates around the $x$-axis, you use $dx$</p>
<p>If it rotates around the $y$-axis, you use the inverse function and calculate
the values of the function to get the limits of the integral.</p>
|
1,144,488 | <p>I was assigned the following differential equation with initial value condition:</p>
<p>$$dy/dx=xy^{1/2}, y(2)=1.$$</p>
<p>Then I found out that the following functions, $y(x)=x^4/16$ and $(x^2-8)^2/16$, are solutions to this equation.</p>
<p>Why does not this contradict the uniqueness of the Picard-Lindelöf theo... | Thomas Andrews | 7,933 | <p>$y(x)=(x^2-8)^2/16$ does not have the property that $y^{1/2}=(x^2-8)/4$, because that is a negative value when $x$ is near $2$.</p>
<p>So $y'=x(x^2-8)/4 = -xy^{1/2}$ when $x\in (0,2\sqrt{2})$, for example.</p>
|
1,144,488 | <p>I was assigned the following differential equation with initial value condition:</p>
<p>$$dy/dx=xy^{1/2}, y(2)=1.$$</p>
<p>Then I found out that the following functions, $y(x)=x^4/16$ and $(x^2-8)^2/16$, are solutions to this equation.</p>
<p>Why does not this contradict the uniqueness of the Picard-Lindelöf theo... | Gonçalo Morais | 226,681 | <p>Or in other words, $y(x)=(x-8)^2/16$ is not a solution of the differential equation. Indeed it would be the solution that corresponds to the initial condition $y(0)=4$. However, by the slope field you see that this solution can not descend to pass through the point $(1,2)$.<img src="https://i.stack.imgur.com/OtB7B.p... |
2,136,183 | <p>Find integers x,y such that the repeating decimal 0.712341234.... = x/y.</p>
<p>I would actually do this problem if the 7 was not there. If the 7 was not there, my proof would be as follows.</p>
<p>proof:</p>
<p>Let z = 0.12341234...</p>
<p>Then 10^4z = 1234.1234</p>
<p>10^4z - z = 1234</p>
<p>z = 1234/(10^4 -... | celtschk | 34,930 | <p>We have
\begin{align}
0.7\overline{1234}
&= \frac{7.\overline{1234}}{10}\\
&= \frac{1}{10}\left(7+0.\overline{1234}\right)\\
&= \frac{1}{10}\left(7+\frac{1234}{9999}\right)\\
&= \frac{1}{10}\left(\frac{7\cdot 9999}{9999} + \frac{1234}{9999}\right)\\
&= \frac{7\cdot 9999 + 1234}{10\cdot 9999}\\
&a... |
2,797,419 | <p>Calculating the moment of inertia is fairly simple, however, how would I proceed to calculate it about an arbitrary axis?
The question asks the moment of inertia of $C=\{(x,y,z)|0\leq x,0\leq y, 0 \leq z,x+y+z\leq 1\}$, so, if I'm not wrong about the boundaries, the moment of inertia about the "usual" Z axis would b... | G Cab | 317,234 | <p>a) Also the variable $z$ shall be limited, presumably $0 \le z$, otherwise the solid is infinite in that direction. </p>
<p>b) Assuming the above, then $C=\{(x,y,z)|0\leq x,0\leq y,0 \le z,x+y+z\leq 1\}$ is a right Tetrahedron,
with edges on the axes and on the diagonal plane $x+y+z=1$.</p>
<p>c) The axis $(t,t,t)... |
1,607,108 | <p>Using the ratio test:
$$\frac{1}{3}\lim\limits_{n\to\infty}\left|\frac{(n+1)^3(\sqrt{2}+(-1)^{n+1})^{n+1}}{n^3(\sqrt{2}+(-1)^n)^n}\right|$$</p>
<p>Without evaluating the limit, the numerator is greater then denominator and the series is divergent. Is there an easier method for checking the convergence of this seque... | Bernard | 202,857 | <p>If $x$ is even, $x^2-1\not\equiv 0\mod 2$. Hence from the hypothesis you know $x$ is odd. Furthermore, the square of any odd integer is congruent to $1$ modulo $8$.</p>
|
2,822,176 | <p>I'm solving a list of exercises of double integrals and they normally have a range for $x$ and $y$, but in this case it says that $y = x^2$ and $y = 4$, so I thought that $x$ would be $\sqrt{y}$, but my answer was wrong. The answer should be 25,60).</p>
<p>A thin metal plate occupies a shadow below the figure below... | Tony Hellmuth | 564,837 | <p>Well there are 2 approaches.</p>
<p>Firstly see the integral you want to compute is simply a box minus the area underneath the curve $y=x^2$. That is easily done by finding the intersection points.</p>
<p>OR one may flip the curve $y=x^2$ and then simply find the area under the new curve $y=-x^2+4$ and the $x$-axi... |
1,413,955 | <p>I get confused when I put the following three notes together:</p>
<ol>
<li>Power set of any set is a $\sigma$-algebra.</li>
<li>If $X$ is a set and $\Sigma$ is a $\sigma$-algebra over $X$, then the pair $(X, \Sigma)$ is a measurable space.</li>
<li><a href="https://en.wikipedia.org/wiki/Vitali_set" rel="nofollow">V... | Zev Chonoles | 264 | <p>It is an unfortunate choice of terminology.</p>
<ul>
<li><p>The term <strong><em>measureable space</em></strong>, taken as a single unit, is <strong>defined</strong> to mean a set $X$ with a chosen sigma-algebra $\Sigma$. There is no specific choice of measure involved, and saying that $(X,\Sigma)$ is a measurable ... |
2,118,230 | <blockquote>
<p>Let $G$ be a group and let $H,K \le G$ be subgroups of finite index, say $|G:H| = m$ and $|G:K|=n$. Prove that $\mathrm{lcm}(m,n) \le |G:H \cap K| \le mn$. </p>
</blockquote>
<p>I was able to establish the upper bound, but I am having difficulty establishing the lower bound, so I consulted <a href="h... | Cross Ratio | 409,170 | <p>If $ \prod p^{m_p} $ and $ \prod p^{n_p} $ are the prime factorizations of $ m $ and $ n $, then their LCM is $ w = \prod p^{max({m_p}, {n_p})} $.
(<a href="https://en.wikipedia.org/wiki/Least_common_multiple#Fundamental_theorem_of_arithmetic" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Least_common_mul... |
588,228 | <p>A,B,C are distinct digits of a three digit number such that </p>
<pre><code> A B C
B A C
+ C A B
____________
A B B C
</code></pre>
<p>Find the value of A+B+C.</p>
<p>a) 16 b) 17 c) 18 d) 19</p>
<... | Brian M. Scott | 12,042 | <p>The largest possible carry from a sum of three digits is $2$, so from the righthand column we see that $2C+B$ is $C$, $10+C$, or $20+C$, and therefore $C+B$ is $0$, $10$, or $20$. The sum of two distinct digits cannot be $0$ or $20$, so $C+B=10$, and there is a carry of $1$ to the middle column.</p>
<p>From the mid... |
1,937,515 | <p>If a function is analytic on $B(0,1)$ and continuous as well as real valued on the boundary (i.e. $|z| = 1$),then show that the function is constant.</p>
| H. H. Rugh | 355,946 | <p>You may also use Schwarz' reflection principle and define for $|z|>1$:
$$ f(z)= \overline{f\left(\frac{1}{\overline{z}} \right)} $$
You may show that this extends $f$ to a uniformly bounded holomorphic function and apply Liouville.</p>
|
3,622,574 | <blockquote>
<p><span class="math-container">$y''-y=\sin2t+1, y(0) = 1, y'(0) = -1$</span></p>
</blockquote>
<p>Hey all, I am not sure if I am completing all the steps correctly to solve this problem via convolution. I am pretty sure I am not doing it correctly, but am not sure what is going wrong. </p>
<p>First I... | Mnifldz | 210,719 | <p>You computed your initial Laplace transform incorrectly. Notice that </p>
<p><span class="math-container">\begin{eqnarray*}
\mathcal{L}[y''] - \mathcal{L}[y] & = & s^2\mathcal{L}[y] - y(0) - y'(0) - \mathcal{L}[y] \\
& = & (s^2-1)\mathcal{L}[y] - 1 - (-1) \;\; =\;\; (s^2-1)\mathcal{L}[y].
\end{eqna... |
3,622,574 | <blockquote>
<p><span class="math-container">$y''-y=\sin2t+1, y(0) = 1, y'(0) = -1$</span></p>
</blockquote>
<p>Hey all, I am not sure if I am completing all the steps correctly to solve this problem via convolution. I am pretty sure I am not doing it correctly, but am not sure what is going wrong. </p>
<p>First I... | user577215664 | 475,762 | <p><span class="math-container">$$y''-y=\sin2t+1, y(0) = 1, y'(0) = -1$$</span>
I got this:
<span class="math-container">$$Y(s) = \left(\frac{2}{s^2+4} + \frac{1}{s} \right) \left(\frac{1}{s^2-1} \right)+\dfrac 1 {s+1}$$</span>
Which is decomposed as:
<span class="math-container">$$Y(s) = -\frac{2}{5(s^2+4)} + \frac 1... |
2,087,199 | <p>Ok so my question is how do you proceed when you need to find the image representation if the basis is a lower RX than your linear transformation.</p>
<p>For example:
Find the image representation of the linear transformation f(x,y,z,t)=(x+y,z-x,t+y) fron $\Bbb R^4$ to $\Bbb R^3$ with respect to the following bases... | spaceisdarkgreen | 397,125 | <p>I think you're misunderstanding. They want you to do it using that weird basis for R4 and the standard basis for R3, which is what you did. This isn't two separate problems. Any problem requires you specify the basis for each space.</p>
|
2,087,199 | <p>Ok so my question is how do you proceed when you need to find the image representation if the basis is a lower RX than your linear transformation.</p>
<p>For example:
Find the image representation of the linear transformation f(x,y,z,t)=(x+y,z-x,t+y) fron $\Bbb R^4$ to $\Bbb R^3$ with respect to the following bases... | Andrea Mori | 688 | <p>Given a linear transformation $T:\Bbb R^n\rightarrow\Bbb R^m$ and a choice of bases $\{e_1,...,e_n\}$ of $\Bbb R^n$ and $\{f_1,...,f_m\}$ of $\Bbb R^m$ is associated a matrix $A$ with $m$ rows and $n$ columns as follows.
You write
$$
A=\left(\begin{array}[cccc]
{} a_{1,1}& a_{1,2}&\cdots& a_{1,n}\\
{} a_... |
1,600,266 | <p>I'm trying to work out sum of this series $$1 + \frac{2}{2} + \frac{3}{2^2} + \frac{4}{2^3} + \ldots$$</p>
<p>I know one method is to do substitutions and getting the series into a form of a known series. So far I've converted the series into $$ 1 + \frac{2}{x} + \frac{3}{x^2} + \frac{4}{x^3} + \ldots $$ where $x=... | J.-E. Pin | 89,374 | <p>There is no mistake and the meaning is clear. However, it might have been simpler to state that $0011101$ is not the binary representation of <strong>any number</strong> (prime or not) because except for $0$, the binary representation of a number always start by $1$.</p>
<p>Let me quote the whole paragraph:</p>
<b... |
2,711,785 | <p>Having some issues with this proof. Assume we've already proven addition, etc.</p>
<p>Definition of multiplication:</p>
<p>$a \times S(b) = a \times b + a$ (the "definition of multiplication")</p>
<p>$a \times 0 = 0$ (the "zero property of multiplication")</p>
<p>First, some supporting proofs:</p>
<p><strong>Cl... | user525966 | 525,966 | <p>I think it's fixed now:</p>
<p><strong>Claim</strong>: $S(a) \times b = a \times b + b$</p>
<p><strong>Base Case</strong>: We induct on $b$. Let $b=0$. See that $S(a) \times 0 = 0 = a \times 0 + 0$ by zero property of multiplication and additive identity.</p>
<p><strong>Inductive Step</strong>: Suppose $S(a) \tim... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.