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 |
|---|---|---|---|---|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| user02138 | 2,720 | <p><span class="math-container">\begin{eqnarray}
1^{3} + 2^{3} + 2^{3} + 2^{3} + 4^{3} + 4^{3} + 4^{3} + 8^{3} = (1 + 2 + 2 + 2 + 4 + 4 + 4 + 8)^{2}
\end{eqnarray}</span>
More generally, let <span class="math-container">$D_{k} = \{ d\}$</span> be the set of unitary divisors of a positive integer <span class="math-conta... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Hans Lundmark | 1,242 | <p>$$\frac{1}{\sin(2\pi/7)} + \frac{1}{\sin(3\pi/7)} = \frac{1}{\sin(\pi/7)}$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Wok | 2,380 | <p><img src="https://i.stack.imgur.com/zBPEc.png" alt="Parallelogram"></p>
<p>$$\left|z+z'\right|^{2}+\left|z-z'\right|^{2}=2\times\left(\left|z\right|^{2}+\left|z'\right|^{2}\right)$$</p>
<blockquote>
<p>The sum of the squares of the sides
equals the sum of the squares of the
diagonals.</p>
</blockquote>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Fred Daniel Kline | 28,555 | <p>By excluding the first two primes, Euler's Prime Product becomes a square:</p>
<p>$$\prod _{n=3}^{\infty } \frac{1}{1-\frac{1}{(p_n)^{2}}}=\frac{\pi ^2}{9}$$</p>
<p>By using multiples of the product of the first two primes, we get the square root:</p>
<p>$$\prod _{n=1}^{\infty } \frac{1}{1-\frac{1}{(n p_1 p_2)^{2... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Rick Decker | 36,993 | <p>$$
\frac{e}{2} = \left(\frac{2}{1}\right)^{1/2}\left(\frac{2\cdot 4}{3\cdot 3}\right)^{1/4}\left(\frac{4\cdot 6\cdot 6\cdot 8}{5\cdot 5\cdot 7\cdot 7}\right)^{1/8}\left(\frac{8\cdot 10\cdot 10\cdot 12\cdot 12\cdot 14\cdot 14\cdot 16}{9\cdot 9\cdot 11\cdot 11\cdot 13\cdot 13\cdot 15\cdot 15}\right)^{1/16}\cdots
$$
[N... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Elias Costa | 19,266 | <p>We have by <a href="http://en.wikipedia.org/wiki/Determinant#Block_matrices">block partition rule for determinant</a>
$$
\det
\left[
\begin{array}{cc}
U & R \\
L & D
\end{array}
\right]
=
\det U\cdot \det ( D-LU^{-1}R)
$$
But if $U,R,L$ and $D$ commute we have that
$$
\det
\left[
\begin{array}{cc}
U &am... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| user45926 | 45,926 | <p>$(x-a)(x-b)(x-c)\ldots(x-z) = 0$</p>
|
4,388,974 | <p>Let <span class="math-container">$f$</span> be a analytic function in the closed unit circle with its center
at the point <span class="math-container">$\alpha\in\mathbb{R}$</span>, then:
<span class="math-container">\begin{equation*}
\int_0^\pi\frac{f\left(\alpha+e^{ix}\right)+f\left(\alpha+e^{-ix}\right)}{1+2p\cos(... | Random Variable | 16,033 | <p>If we convert the integral into a contour integral, we get</p>
<p><span class="math-container">$$ \begin{align} \int_{0}^{\pi} \frac{f(\alpha+e^{ix})+f(\alpha+e^{-ix})}{1+2p \cos x +p^{2}} \, \mathrm dx &= \frac{1}{2}\int_{-\pi}^{\pi} \frac{f(\alpha+e^{ix})+f(\alpha+e^{-ix})}{1+2p \cos x +p^{2}} \, \mathrm dx ... |
495,069 | <p>Find an equation of the plane.
The plane that passes through the line of intersection of the planes
$x − z = 1$ and $y + 4z = 1$
and is perpendicular to the plane
$x + y − 2z = 2$.</p>
<p>I keep getting the answer of $7x-y+5z=6$ and I am told that it is wrong. I do not understand what I am doing wrong.
I have th... | Sharabh | 95,079 | <p>This is actually a 2 line problem which can be solved as follows:
Equation of the plane passing through the planes x−z-1=0 and y+4z-1=0 can be written as
x−z-1 + k(y+4z-1) = 0 for some constant k (which is required to be found out).
i.e., x + ky + (4k-1)z - k - 1 = 0 .... (1)
Now since this plane is perpendicular to... |
314,239 | <blockquote>
<p>If <span class="math-container">$P_1 , P_2 $</span> are two sylow <span class="math-container">$p$</span>-subgroups of the group <span class="math-container">$G$</span> prove that:</p>
<p><span class="math-container">$ P_1 \bigcap $</span> <span class="math-container">$P_2$</span> = <span class="math-co... | Jyrki Lahtonen | 11,619 | <p>This is not true in general. For example the group $S_6$ has the following two Sylow 2-subgroups. Let $P_1$ be generated by $(12),(13)(24)$ and $(56)$ (the first two permutations generate a Sylow 2-subgroup of $S_4$). Let $P_2$ be generated by $(12),(35)(46)$ and $(56)$ (here we have a Sylow 2-subgroup of $S_4$ in a... |
1,310,530 | <p>From: $2015$ Singapore Mathematical Olympiad Secondary 2 (Grade 8) Question 21 Round 1 on 3rd June.</p>
<blockquote>
<p>Find the value of $\sqrt {(98 \times 100+2)(100\times102+2)+(100\times2)^2}$ (No use of calculators)</p>
</blockquote>
<p>My attempt:Special cases formula -> $x^2=(x+1)(x-1)+1$</p>
<p>Therefor... | Mark Bennet | 2,906 | <p>I would do something like you have done, but just put $x=100$ straight away since that is the central number. You can always do the $x^2-1$ simplification afterwards, if it helps, but I would get rid of the big numbers as quickly as possible.</p>
<p>Then $$\left(x(x+2)+2\right)\left(x(x-2)+2\right)+4x^2=x^2(x^2-4)+... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Denis Serre | 8,799 | <p>Jordan algebras were introduced first by P. Jordan and J. von Neumann in order to give a mathematical context for observables in quantum mechanics (say, a structure that generalizes the space of Hermitian matrices). At the end, the classification was disappointing, and Jordan algebras do not play a role any more in ... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | JSE | 431 | <p>"How much would you subscribe to the statement that studying questions one finds interesting is something established mathematicians do, while younger ones are better off studying questions that the rest of the community finds interesting?"</p>
<p>Not at all. I don't think anyone, young or old, will find ... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Scott Carter | 36,108 | <p>I am thinking of specific examples. In much the same way, David Corfield mentioned groupoids. </p>
<p>I am personally not a big fan of the general theory of loops. In part, my own disinterest is because I have not found an application. On the other hand, I have seen enough to believe that Moufang loops are interest... |
1,729,893 | <p>Let $\nu_1, \nu_2, \nu_3 \in \mathbb{R}$ not all be zero.</p>
<p>I wish to show
$$\nu_1^2 + \nu_1\nu_2 + \nu_2^2+\nu_2\nu_3 + \nu_3^2 > 0\text{.}$$</p>
<p><a href="http://www.wolframalpha.com/input/?i=x%5E2%2Bx*y%2By%5E2%2By*z%2Bz%5E2+%3E+0" rel="nofollow">Wolfram</a> seems to suggest splitting this into cases,... | Ted Shifrin | 71,348 | <p><strong>HINT</strong>: Complete the square. So start with $(v_1+\frac12v_2+\frac12v_3)^2$.</p>
|
239,900 | <p>Hatcher states the following theorem on page 114 of his Algebraic Topology:</p>
<blockquote>
<p>If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence
$$...\longrightarrow\widetilde{H}_n(A)\overset{i_*}\longrightarrow \w... | apurv | 49,824 | <p>One example which occurs a lot "in nature" is when $A$ is a large open subset of $X$. For example, $X$ is a manifold and $A = X - \text{point}$ or when $X$ is the total space of a vector bundle and $A$ is the complement of the <strike>base space</strike> zero section. However, in these cases, it is often possible to... |
2,125,297 | <p><a href="https://i.stack.imgur.com/cBCJe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cBCJe.png" alt="enter image description here"></a></p>
<p>I would like to know how exactly this works. I watched a khan academy video: "Multiplying matrices" but in this case he would've done B*A and had 2 co... | kotomord | 382,886 | <p>Square of the distance from (1,1,1,1) to W is minimum of </p>
<p>$(1-x)^2 + (1+x)^2 +(1-y)^2 + (1-y)^2 = 2+ 2x^2 + (1-y)^2$ over $\mathbb{R}^2$</p>
<p>Simple to see it is 2 </p>
|
698,474 | <p>I am trying to find the phi(18). Using an online calculator, it says it is 6 but im getting four.
<br/>
The method I am using is by breaking 18 down into primes and then multiplying the phi(primes)</p>
<p>$$=\varphi (18)$$
$$=\varphi (3) \cdot \varphi(3) \cdot \varphi(2)$$
$$= 2 \cdot 2 \cdot 1$$
$$= 4$$</p>
| NasuSama | 67,036 | <p>Remember that you need to determine the prime factorization of $18$. That is, $18 = 3^2 \cdot 2$. Since $18 = 3^2 \cdot 2$, we have</p>
<p>$$\begin{aligned}
\varphi(18) = \varphi(3^2) \cdot \varphi(2) &= (3^2 - 3)(2 - 1) = 6
\end{aligned}$$</p>
<p>So in general, if $k = p_1^{n_1}p_2^{n_2}\cdots p_s^{n_s}$, t... |
1,243,750 | <p>$A \in \mathbb{R}^{n\times n}$, with $A^2 = 1$ and $A\ne\pm1$</p>
<p>Show that the only eigenvalues of $A$ are $1$ and $-1$.</p>
| Thomas Russell | 32,374 | <p>We have that:</p>
<p>$$\mathbf{A}^{2} = \mathbf{I} \implies \mathbf{A} = \mathbf{A}^{-1}$$</p>
<p>But we know that if the eigenvalues of $\mathbf{A}$ are $\lambda_{1},\dots,\lambda_{n}$, then the eigenvalues of $\mathbf{A}^{-1}$ are $\lambda_{1}^{-1},\dots,\lambda_{n}^{-1}$, but if:</p>
<p>$$\lambda_{i}=\lambda_{... |
587,484 | <p>Show that $\left(\displaystyle\frac{\partial (u,v)}{\partial (x,y)}\right)\left(\displaystyle\frac{\partial (x,y)}{\partial (r,s)}\right)=\displaystyle\frac{\partial (u,v)}{\partial (r,s)}$. Thus, prove that $\displaystyle\frac{\partial (u,v)}{\partial (x,y)}=\frac{1}{\displaystyle\frac{\partial (x,y)}{\partial (u,v... | user110373 | 110,373 | <p>Lemma: Let $A, B, C$ be $m \times m $ matrices such that $A \times B= C$, then $detC= detA \cdot detB$. </p>
<p>Proof: By lemma, it's sufficient to show that $$ \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y} \end{... |
105,058 | <p>Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, $S$ of integers.</p>
<p>When $S$ is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which ... | J.. | 4,743 | <p>No. Such a set $S$ must include the primes (because they have no other factors). If $s \in S$ is not prime, then it can be written as a product of primes, i.e., as a product of other elements of $S$. This contradicts uniqueness. Therefore the only such set $S$ is the set of the primes.</p>
|
105,058 | <p>Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, $S$ of integers.</p>
<p>When $S$ is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which ... | Steven Stadnicki | 785 | <p>When you relax the integrality condition you start to pick up other solutions - for instance, the set of square roots of primes, $S=\{\sqrt{p}:p\in P\}$, is such that every number $n$ has a unique factorization $n=s_1^{e_1}s_2^{e_2}\ldots$ in terms of the elements of $S$; just consider the breakdown $n=p_1^{d_1}p_2^... |
2,378,717 | <p>I wondered many times how the editors of mathematical journals know that the content of a paper or article that was submited by an author is original, this is, that the main content of such calculations and reasonings is unpublished.</p>
<blockquote>
<p><strong>Question.</strong> How do editors of mathematical jo... | Tobias Kildetoft | 2,538 | <p>When submitting a paper for publication, it is not only the editor who looks at it. The editor will (usually) take a first look to determine if the paper looks to be suitable for the journal. If they see that the main results are already well-known this will likely make them reject the paper (especially if the autho... |
2,665,539 | <p>How to solve the following system of equations</p>
<blockquote>
<p>$$
\begin{cases}
a+c=12\\
b+ac+d=86\\
bc+ad=300\\
bd=625\\
\end{cases}
$$</p>
</blockquote>
| Dave | 334,366 | <p>This system of equations corresponds to a factorization of $$f(x)=x^4+12x^3+86x^2+300x+625$$ into two quadratics as
$$f(x)=(x^2+ax+b)(x^2+cx+d)$$
The roots of $f(x)$ are $-3-4i$ and $-3+4i$, each having multiplicity $2$. So what should the quadratic factors of $f(x)$ look like?</p>
|
331,169 | <p>Let $\Omega\subset \mathbb{R}^n$ be bounded and let $X:=H^1(\Omega)$. Let $a:\Omega\times \mathbb{R} \to \mathbb{R}, (x,z)\mapsto a(x,z)$ be a bounded function such that $a(x,.)$ is continuous on $\mathbb{R}$ for every $x$ and $a(.,z)$ is measurable on $\Omega$ for every $z$.</p>
<p>For $v,\phi\in H^1$ one finds in... | Julien | 38,053 | <p>Take an element $(n,m)\in\mathbb{Z}^2=\mathbb{Z}\times\mathbb{Z}$. If it is a generator, then $(1,0)$ and $(0,1)$ are in $\mathbb{Z}\cdot(n,m)$ the group generated by $(n,m)$. Write down what this actually means. Can you reach a contradiction?</p>
|
1,843,724 | <p>I know that the Maclaurin expansion of $e^x$ is $$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$
But i'm not sure how to find the Maclaurin series here
I tried this</p>
<p>$$
f'_{(0)}=-2xe^{-x^2}=0
$$
And that follows to every derivative that follows, so how can I get a power series out of it?</p>
| Daniel | 150,142 | <p>As Dark mentions in the comments,
$$e^y = 1+y+\frac{y^2}{2!}+\frac{y^3}{3!}+\cdots$$
works for <strong>any</strong> $y\in \Bbb R$. In particular, you can substitute $y=-x^2$ and that should give you the result!</p>
<p>$$e^{-x^2} = 1+\left(-x^2\right)+\frac{(-x^2)^2}{2!}+\frac{(-x^2)^3}{3!}+\cdots$$</p>
<p>There's ... |
3,290,073 | <p>I'm trying to help my daughter learn maths. She is struggling with factors, which is to work out what numbers go into a larger number (division).</p>
<p>I've already learned that by summing numbers, if they make 3, it can divide by 3. I also know the rules for 2, 5, 6, 9 and 10.</p>
<p>I'm trying to see if there i... | J. W. Tanner | 615,567 | <p>How to make sense of that rule for divisibility by <span class="math-container">$4$</span>:
it's not saying to <em>add</em> the last two digits; it's merely saying to <em>look</em> at the last two digits. Because <span class="math-container">$4$</span> divides <span class="math-container">$100$</span>, a number is ... |
2,086,134 | <p>Solve for $x$ if $4^x + x^x = 20$ where $x \in \mathbb{Z}$ </p>
<p>My Attempt: </p>
<p>By inspection,<br>
$x=2$</p>
<p>Now,
$4^2 + 2^2 = 20$</p>
<p>$16+4=20$</p>
<p>$20=20$</p>
<p>Which is True.</p>
<p>But, I Could not do it by any process. Can anyone help me? </p>
<p>Thanks in Advance.</p>
| projectilemotion | 323,432 | <p>There does not exist an explicit solution for $x$ which can be expressed in a finite number of elementary functions.</p>
<p>However, you can solve this using an <a href="https://en.wikipedia.org/wiki/Iterative_method" rel="nofollow noreferrer">iterative method</a>. In order to apply it, we will generalise our possi... |
225,481 | <p>I would like to write a program that gives me the Klein Gordon Equations given a metric. I will explain.</p>
<p>My code is in the following:</p>
<p><strong>I) Standard Quantities</strong></p>
<p>I have no doubt here the first part is <a href="http://web.physics.ucsb.edu/%7Egravitybook/mathematica.html" rel="nofollow... | xzczd | 1,871 | <p>You can try my <a href="https://mathematica.stackexchange.com/a/216806/1871"><code>allowtensor</code></a>. Since it's a 4D problem, you need to set the value of global variable <code>$tensordimension</code> to <code>4</code>:</p>
<pre><code>dmetric = Det[metric];
ref = KG00 + KG11 + KG22 + KG33;
$tensordimension = 4... |
3,096,933 | <p>In studying the causes of power failures, the following data have been gathered:
10% are due to a transformer damage, 75% are due to line damage, 5% involve both
problems. Based on these percentages, find the probability that a given power failure
involves:</p>
<p>a) line damage given that there is transformer dama... | user0102 | 322,814 | <p>According to your notation, we have <span class="math-container">$\textbf{P}(T) = 0.1$</span>, <span class="math-container">$\textbf{P}(L) = 0.75$</span> and <span class="math-container">$\textbf{P}(T\cap L) = 0.05$</span>. Therefore:</p>
<p>(a) The sought probability is given by <span class="math-container">$\text... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | Omran Kouba | 140,450 | <p>This might be a possible solution. For a positive integer $n$, let $\nu_2(n)$ be the largest $k$ such that $2^k|n$, and similarly, let $\nu_3(n)$ be the largest $k$ such that $3^k|n$. Finally let
$$h(n)=\frac{n}{3^{\nu_3(n)}2^{1+4\lfloor \nu_2(n)/4\rfloor}}$$
If we consider
$$
a\times ~ b {\buildrel \rm def\over =}~... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | Asinomás | 33,907 | <p>Here is something I did which lead me to an incorrect result, but it is still pretty close.</p>
<p>Since all the values we are given are of the form $a\times (a+1)$, I decided to make the function $f(a)=a\times (a+1)$. Assuming $f$ is a polynomial of grade $4$ or less we obtain $f$ is equal to $\frac{101 x^3}{6}-23... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | vuur | 85,875 | <p>One solution is to define the operation $\times$ between two integers as</p>
<p>$m \times n = n \cdot
\left\{
\begin{array}{ll}
\frac{1}{3}\sum_{k=1}^m k &\mbox{if } 3 \mid \sum_{k=1}^m k \\
\sum_{k=1}^m k &\mbox{otherwise.}
\end{array}
\right.$</p>
<p>The point is, that what remains of the RHS after... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | user2477732 | 164,475 | <p>numbers sequence</p>
<p>3 . 4 = 4 X 2 = 8</p>
<p>4 . 5 = 5 X 10 = 50</p>
<p>5 . 6 = 6 X 5 = 30</p>
<p>6 . 7 = 7 X 7 = 49</p>
<p>7 . 8 = 8 X 28 = 224</p>
<p>8 . 9 = 9 X 4 = 36</p>
<p>9 . 10 = 10 X 5 = 50... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | I. J. Kennedy | 130 | <p>Surely even this interviewer meant for <span class="math-container">$\times$</span> to be commutative, so I propose
<span class="math-container">$$a \times b = \left( \binom{a+1}{3},\binom{b+1}{3} \right) \cdot \max(a,b)$$</span>
where the inner brackets are binomial coefficients, and the outer brackets indicate the... |
1,401,760 | <p>First I tried to use integration:
$$y=\lim_{n\to\infty}\frac{a^n}{n!}=\lim_{n\to\infty}\frac{a}{1}\cdot\frac{a}{2}\cdot\frac{a}{3}\cdots\frac{a}{n}$$
$$\log y=\lim_{n\to\infty}\sum_{r=1}^n\log\frac{a}{r}$$
But I could not express it as a <em>riemann integral</em>. Now I am thinking about sandwich theorem.</p>
<p>$$... | Zach466920 | 219,489 | <p>What is,</p>
<p>$$\lim_{n \to {\infty}} {{a^n} \over {n!}}$$</p>
<p>Well, for large values of $n$, $n!$ can be evaluated with <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow">Stirling's Approximation</a>.</p>
<p>$$\lim_{n \to {\infty}} {{a^n} \over {n!}}=\lim_{n \to {\infty}} {{a^... |
1,401,760 | <p>First I tried to use integration:
$$y=\lim_{n\to\infty}\frac{a^n}{n!}=\lim_{n\to\infty}\frac{a}{1}\cdot\frac{a}{2}\cdot\frac{a}{3}\cdots\frac{a}{n}$$
$$\log y=\lim_{n\to\infty}\sum_{r=1}^n\log\frac{a}{r}$$
But I could not express it as a <em>riemann integral</em>. Now I am thinking about sandwich theorem.</p>
<p>$$... | idm | 167,226 | <p>Let $x_n=\frac{a^n}{n!}$.</p>
<p>$$\left|\frac{x_{n+1}}{x_n}\right|=\frac{\frac{a^{n+1}}{(n+1)!}}{\frac{a^n}{n!}}=\frac{a^{n+1}n!}{a^n (n+1)!}= \frac{a}{n+1}\underset{n\to \infty }{\longrightarrow }0$$</p>
<p>and thus, by $x_n\to 0$ by <a href="https://en.wikipedia.org/wiki/Ratio_test" rel="nofollow">Ratio test</a... |
3,635,266 | <p>As stated in title, I am curious how to construct a sequnce <span class="math-container">$\{a_n\}_{n=1}^{\infty}$</span> such that <span class="math-container">$\sum_{n=1}^{\infty} a_n^2 < +\infty$</span> but <span class="math-container">$\sum_{n=1}^{\infty}\frac{a_n}{\sqrt{n}} = \infty$</span>.</p>
<p>This prob... | QC_QAOA | 364,346 | <p>Consider</p>
<p><span class="math-container">$$a_n=\frac{1}{\sqrt{n}\ln(n+1)}$$</span></p>
<p>Then</p>
<p><span class="math-container">$$\sum_{n=1}^\infty a_n^2=\sum_{n=1}^\infty\frac{1}{n\ln(n+1)^2}<\infty$$</span></p>
<p>but</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{a_n}{\sqrt{n}}=\sum_... |
63,641 | <p>Here is the problem: Two mathematicians meet at a bar. They like each other and tend to collaborate. But it is not so clear what problems could be of common interest to both of them. Of course, the traditional way is they keep describing they work or their field in general so that hopefully they catch something at t... | GH from MO | 11,919 | <p>The Mathieu group $M_{11}$ does not have this property. A quote from Example 2.16 in <a href="http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3895v1.pdf">this paper</a>: "Hence there is no automorphism of $M_{11}$ that maps $x$ to $x^{−1}$."</p>
<p>Background how I found this quote as I am no group theorist: I used G... |
642,497 | <p>How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? </p>
<p>(taking $ABCD$ to be the trapezium with
$AD$ || $BC$ and $O$ the intersection point of diagonals , if it can be shown by vector method that
$OB=OC$ , ... | egreg | 62,967 | <p>Hint: Set $y=x+2$, so that $x=y-2$; do your math and …</p>
<p>Hint 2: Taylor expansion.</p>
|
398,002 | <p>There is a <a href="https://en.wikipedia.org/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" rel="noreferrer">classical correspondence</a> between Lie algebras (over <span class="math-container">$\mathbb{R}$</span> or <span class="math-container">$\mathbb{C}$</span>) and Lie groups in the finite dimensional case:... | Robert Bryant | 13,972 | <p>Here is an informative example that illustrates the difficulties: Consider the Lie algebra <span class="math-container">${\frak{g}} = \mathrm{Vect}(\mathbb{S})$</span> of smooth vector fields on the circle <span class="math-container">$\mathbb{S}$</span>. The flow of any vector field <span class="math-container">$... |
2,307,268 | <p>I tried to compute the power of 2^2^2^2 on google calculator and my casio calculator but both are giving different results. same is true for 3^3^3.
Please explain me the difference between two expressions.</p>
<p><a href="https://i.stack.imgur.com/2CMDt.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | John | 7,163 | <p>The difference lies in the applied order of operations.</p>
<p>The Casio calculator does operations strictly left to right unless you break it with parentheses:</p>
<blockquote>
<p>2^2^2^2 = 4^2^2 = 16^2 = 256</p>
</blockquote>
<p>But Google evaluates the entire expression (correctly) using right-to-left order ... |
1,201,352 | <p>I have a very simple question which is bugging me.</p>
<p>We are 3 roommates and our total electricity bill is $61 this month,
I was home for the whole month,
Friend X was home for 15 days,
Friend Y was home for 20 days</p>
<p>now the easy question, how much would each person would pay?</p>
<p>My calculation is <... | Community | -1 | <p>n = $61/(30.5 + 15 + 20)</p>
<p>30.5 X n = $28.40</p>
<p>15 X n = $13.97</p>
<p>20 X n = $18.63</p>
<p>The individual amounts total $61.</p>
|
1,920,781 | <p>Let's say I have a basic equation (here a linear combination):
$$ax_1 + bx_2=y$$</p>
<p>Is there any way to get the value of the sum $x_1+x_2$ if $a$, $b$ and $y$ is known?</p>
| MPW | 113,214 | <p>In the absence of any additional information, no.</p>
<p>For example, there are lots of pairs of numbers that differ by $1$, but their sum could be anything. Consider the pairs (0,1), (10,11), (1000, 1001).</p>
|
3,041,841 | <p>Let <span class="math-container">$f(z)$</span> be holomorphic function on an open unit disk such that <span class="math-container">$\lim_{z \to 1} f(z)$</span> doesn't exists. Let <span class="math-container">$f(z) = \Sigma_{i=1}^{\infty} a_iz^i$</span> be its Taylor series around <span class="math-container">$0$</s... | nmasanta | 623,924 | <p>Given that <span class="math-container">$f(z)$</span> be holomorphic function on an open unit disk, so
let <span class="math-container">$$f(z)=\frac{1}{1-z}$$</span>
then <span class="math-container">$\quad\lim_{z \to 1} f(z)\quad$</span> does not exist.</p>
<p>Taylor series of <span class="math-container">$f(z)$<... |
3,609,791 | <p><span class="math-container">$f(x) = \frac{x}{(1+x^2)}$</span></p>
<p>I need to find the range of function.</p>
<p>My method-</p>
<p>Let <span class="math-container">$f(x)=y$</span> </p>
<p>Then <span class="math-container">$x²y-x+y=0$</span></p>
<p><span class="math-container">$$x=1\pm\frac{\sqrt{(1-4y²)}}{2y}... | Luca Goldoni Ph.D. | 264,269 | <p>When you have a second degree equation <span class="math-container">$ax^2+bx+c=0$</span> in order to use the formula
<span class="math-container">$$
x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}
{{2a}}
$$</span>
it must be <span class="math-container">$a \neq 0$</span>. As <span class="math-container">$a=0$</span> your... |
2,072,061 | <p>My Working: </p>
<p>\begin{align}
a_0=&-0+2=2\\
a_1=&-1+2=1\\
a_2=&-2+2=0.
\end{align}
and $$a_2=a_{2-1}+2a_{2-2}+2(2)-9=-4.$$</p>
<p>I don't know what's wrong with my solution. I used this method on various questions but now I'm just ending up with the wrong answer.</p>
| Arnaldo | 391,612 | <p>We have:</p>
<p>$$a_n=-n+2$$
$$a_{n-1}=-(n-1)+2=-n+3$$
$$a_{n-2}=-(n-2)+2=-n+4$$</p>
<p>And then </p>
<p>$$a_n-a_{n-1}-2a_{n-2}=-n+2-(-n+3)-2(-n+4)= 2n-9$$</p>
|
812,263 | <p>To further explain the title:</p>
<p>Is there a probabilistic reason as to why a 6-sided die has the opposing sides suming to 7?</p>
<p>My argument begun when a friend decided to use <a href="http://ecx.images-amazon.com/images/I/51OvRGphnOL.jpg" rel="nofollow">this die</a> instead of <a href="http://nerdywithchil... | Tara | 150,046 | <p>In the 20-sided dice linked, there is actually a good reason not to use the die with numbers of equal size clustered closely together.</p>
<p>Just as in a roulette wheel, it is harder to cheat if a slight difference in movement totally changes the number, and in addition, it is much more suspenseful to watch if you... |
3,804,195 | <p>I am currently reworking the following exercise that was worked during my previous class's lecture on normed vector spaces:</p>
<blockquote>
<p>Show that <span class="math-container">$||x||_{\infty} \leq ||x||_{2} \leq ||x||_{1}$</span> for any <span class="math-container">$x \in \mathbb{R}^n$</span>.</p>
</blockquo... | RobPratt | 683,666 | <p>Yes, you can use linear programming to find such <span class="math-container">$p_k$</span>. For each <span class="math-container">$k$</span>, let <span class="math-container">$p_k$</span> be the desired weight. The objective is arbitrary, so you can minimize the constant <span class="math-container">$0$</span> fun... |
2,131,224 | <blockquote>
<p>Say I have vectors $x, y$, then is $\text{proj }_x y $ a scalar multiple of $x$?</p>
</blockquote>
<p>I have a book saying that it is, but I have no clue why this true. Is this really true?</p>
| Martin Argerami | 22,857 | <p>Of course. What else could the projection be? It has to be an element of the subspace spanned by $y$, which consists precisely of the scalar multiples of $x$. </p>
|
457,427 | <p>Find the derivative of the functions:<br>
$$\int_{x^2}^{\sin(x)}\sqrt{1+t^4}dt$$<br></p>
<p>In class we had the following solution:<br>
By the fundamental theorem of calculus we know that <br>
$$\left(\int_a^xf(t)dt\right)'=f(x)$$ So<br>
$$\int_{x^2}^0\sqrt{1+t^4}dt+\int_0^{\sin(x)}\sqrt{1+t^4}dt=$$<br>
$$\int_0^{\... | Mikasa | 8,581 | <p>Generally, the way @Ron walked, is known as <a href="http://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow">Leibniz integral rule</a> which follows from the chain rule:
$$ {d\over dx} \left( \int_{f(x)}^{g(x)} \phi(t) \,dt \right )= \phi\left(g(x)\right) {g'(x)} - \phi(f(x)) {f'(x)} $$</p>
|
33,424 | <p>Suppose $f(z) = P(z)e^{Q(z)}$ where $P,Q$ are real polynomials. What is the number of non-real zeros of $f^{(k)}$ as $k$ increases?</p>
<p>We know that $f''$ has $\geq m$ zeros where $m$ depends on $Q(z)$. </p>
| Pietro Majer | 6,101 | <p>Just some hints. The functions $f^{\,(k)}$ have the same zeros of the polynomials $P_k:=f^{\,(k)}\exp(-Q)$, that satisfy $P_0:=P$ and $P_{k+1}=P_k'+P_kQ'$. In particular $P_k$ has degree $\deg(P)+k\left(\deg(Q)-1\right)$, and this is also the total number of zeros of $f^{\,(k)}$. They may be all real: for instance ... |
2,038,617 | <p>The question asks to show that $$\sum_{k=0}^{3n}(-3)^k \binom{6n}{2k}=2^{6n}$$ by considering the binomial expansion </p>
<p>I thought about the use of $$(1+z)^n=\sum_{k=0}^{n}\binom{n}{k}z^k$$
with suitable complex number $z$, as the formula shows the $(-3)^k$ term might suggest the use of complex number, which ta... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\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}\,}
\... |
2,737,144 | <p>I tried to prove it by contradiction. </p>
<p>Suppose it is not true that $1\ge\frac{3}{x(x-2)}$, so $1\lt\frac{3}{x(x-2)}$. Then $\frac{3}{x(x-2)}-1\gt0$. Multiply both sides of $\frac{3}{x(x-2)}-1\gt0$ by ${x(x-2)}$.</p>
<p>$(\frac{3}{x(x-2)}-1\gt0)({x(x-2)}\gt0(x(x-2)$</p>
<p>${3-(x(x-2)\gt0}$</p>
<p>${... | TheSimpliFire | 471,884 | <p>Note that if $x>3\implies x-2>1$ then $\dfrac1x<\dfrac13$ and $\dfrac1{x-2}<1$ so $$\frac3{x(x-2)}<\frac3{3(3-2)}=1$$</p>
|
2,737,144 | <p>I tried to prove it by contradiction. </p>
<p>Suppose it is not true that $1\ge\frac{3}{x(x-2)}$, so $1\lt\frac{3}{x(x-2)}$. Then $\frac{3}{x(x-2)}-1\gt0$. Multiply both sides of $\frac{3}{x(x-2)}-1\gt0$ by ${x(x-2)}$.</p>
<p>$(\frac{3}{x(x-2)}-1\gt0)({x(x-2)}\gt0(x(x-2)$</p>
<p>${3-(x(x-2)\gt0}$</p>
<p>${... | Bernard | 202,857 | <p>Why do you make things more complex than they are? </p>
<p>The function $x(x-2)$ is <em>increasing</em> and <em>positive</em> on $(2,+\infty)$, hence
$\;\dfrac2{x(x-2)}\;$ is <em>decreasing</em> and <em>positive</em> on this interval.
Furthermore $f(3)=1$…</p>
|
1,679,772 | <p>A binary relation R on $N×N$ is defined as follows$: (a,b)R(c,d)$ if $a≤c$ or $b≤d$. Consider the following
propositions:</p>
<p>$P: R$ is reflexive</p>
<p>$Q: R$ is transitive</p>
<p>Which one of the following statements is TRUE?</p>
<ol>
<li>Both $P$ and $Q$ are true. </li>
<li>$P$ is true and $Q$ is false. </... | Hagen von Eitzen | 39,174 | <p>(I'll use <em>sequence compactness</em> here; for metric spaces this is equivalent to compactness)</p>
<p>Pick $x_n\in K_n$. As $x_n\in K_1$, there is a converging subsequence, so we may assume wlog. that the original sequence is already convergent, say $x_n\to x_\infty$.
For any $i$, we have $x_n\in K_i$ for almos... |
870,367 | <p>Suppose $A$ is a $2$-by-$2$ matrix, and $\mathcal{l}$ is an invariant line under $A$, so $(x,mx+c)$ is mapped to $(X,mX+c)$ for some variable $X$ linear in $x$. Then is there a point on the line $\mathcal{l}$ which is a fixed point of $A$, i.e. there is some $x' \in \mathcal{l}$ such that $Ax'=x'$?</p>
<p>The reaso... | Juanito | 153,015 | <p>As Andre commented, "at least 11" means, in this case, 11 or 12. So, find the the numbers for 11 and 12, and then add them. Answer${}=\dfrac{^{18}C_{12}+{^{18}C_{11}}\cdot10}{^{28}C_{12}}$</p>
|
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | mweiss | 29 | <p>There are possibly two different issues here.</p>
<p>Issue 1 is that some students are under the mistaken impression that the symbol <span class="math-container">$\sqrt{5}$</span> actually designates <em>two different numbers</em>, one positive and one negative. I have even heard high school teachers (in the Unite... |
126,057 | <p>Hi,</p>
<p>Can one define a Fubini-Study metric/Kaehler metric on the projective space of an infinite dimensional Hilbert space, i.e. using the formula $\partial \bar{\partial} \log |Z|^2$?
This should be very well-known to the experts. Anyhow I don't have much experience with infinite dimension and worried that s... | alvarezpaiva | 21,123 | <p>Maybe it's easier to see that the definition extends if you <strong>don't</strong> use a formula:</p>
<p>The unit sphere inherits a Riemannian metric from the Hilbert space in the standard manner and since it is invariant under the circular symmetry $(e^{i\theta},x) \mapsto e^{i\theta} x$, it will
project down to ... |
307,822 | <p>Excuse my incorrect use of terminology, I hope my question is clear:</p>
<p>I am coding a Python module which tests whether a given number is a member of the Fibonacci series. No problem with that. Additionally, should a number not be a member of the series, I would like to test whether it is significantly close to... | Jyrki Lahtonen | 11,619 | <p>The Fibonacci numbers, the larger ones in particular, are very close to being given by the formula
$$
F_n\approx\frac1{\sqrt5} \left(\frac{1+\sqrt5}2\right)^n
$$
($F_n$ is always the closest integer to the r.h.s.).
In light of this I would also consider calculating, given input $x$, how far the ratio
$$
r(x):=\frac... |
3,643,917 | <p>Find the limit of
<span class="math-container">$$\lim_{x\to 0+} \frac{1}{x^2} \int_{0}^{x} t^{1+t} dt$$</span>. </p>
<p>My idea to solve it is to use L'Hospital's rule but I am not sure why I can use it and how should i do it. Many thanks to them who are willing to help. </p>
| Jephph | 777,586 | <p>Let <span class="math-container">$L=\lim_{x \rightarrow 0^+}\frac{1}{x^2}\int_{0}^{x}t^{1+t}dt = \lim_{x \rightarrow 0^+}\frac{\int_{0}^{x}t^{1+t}dt}{x^2}$</span>.</p>
<p>Note that both the numerator and the denominator tend to <span class="math-container">$0$</span> as <span class="math-container">$x$</span> tends... |
197,667 | <p>I have a table with 10 columns, obtained as import from file. I would like to use 4 columns as <code>{x, y, z, u}</code> params for <code>BubbleChart3d</code>, and an additional column to color the points. I'm unable to make this work. The code I have is like this:</p>
<p><code>t</code> is table from a file import.... | Anton.Sakovich | 59,438 | <p>Mathematica does not evaluate limits unless explicitly asked to do so. What it does is simply evaluate the numerator, then evaluate the denominator, and then evaluate the resulting fraction. The only subtlety is that "evaluate the resulting fraction" means a simple syntactic evaluation, without bothering of possible... |
1,574,290 | <p>How do I prove this? </p>
<p>For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.</p>
| Nathan Marianovsky | 257,054 | <ul>
<li>Base Case:</li>
</ul>
<p>For $n = 1$:</p>
<p>$$f_2^2 - f_2f_1 - f_1^2 = (-1)^1$$</p>
<p>$$(1)^2 - (1)(1) - (1)^2 = -1$$</p>
<p>$$-1 = -1$$</p>
<hr>
<ul>
<li>Induction Step:</li>
</ul>
<p>Assume that the given statement is true. We now try to prove that it holds true for $n+1$:</p>
<p>$$f_{n+2}^2 - f_{n... |
2,423,086 | <p>Show, by listing the elements or how we can list the elements, that </p>
<p>$\mathbb{N}^3$ = $\mathbb{N×N×N}$ is a countable set.</p>
<p><strong>Attempt at a solution:</strong></p>
<p>I was thinking about making use of Cantor's diagonal argument. If this were $\mathbb{N×N}$ I could just do:</p>
<p>(1,1), (1,2), ... | Kenny Lau | 328,173 | <p>Let $S = \Bbb N$ and $T = \Bbb N \times \Bbb N \times \Bbb N$.</p>
<p>Then, $f:S \to T$ defined by $f(n) = (n,0,0)$ is an injection, and so is $g:T \to S$ defined by $g(n,m,k) = 2^n 3^m 5^k$.</p>
<p>Hence, there exists a bijection between $S$ and $T$ per <a href="https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%9... |
2,423,086 | <p>Show, by listing the elements or how we can list the elements, that </p>
<p>$\mathbb{N}^3$ = $\mathbb{N×N×N}$ is a countable set.</p>
<p><strong>Attempt at a solution:</strong></p>
<p>I was thinking about making use of Cantor's diagonal argument. If this were $\mathbb{N×N}$ I could just do:</p>
<p>(1,1), (1,2), ... | David K | 139,123 | <p>If you have already found an explicit bijection $f:\mathbb N\to\mathbb N\times\mathbb N$
defined by
$f(n)\mapsto(g(n), h(n)),$
you can extend this to a bijection
$k:\mathbb N\to\mathbb N\times\mathbb N\times\mathbb N$
by setting
$$
k(n)=(g(n), g(h(n)), h(h(n))).
$$
You can recursively apply this technique as many ... |
3,685,969 | <p>I have been investigating the brachistochrone problem with friction and in my derivations, I would like help solving the Euler-Lagrange equation below</p>
<p><span class="math-container">$\frac{d}{dx}\frac{\partial F}{\partial y'}=\frac{\partial F}{\partial y}$</span>
where <span class="math-container">$F=\sqrt{\fr... | Claude Leibovici | 82,404 | <p>I suppose that the problem with Mathematica came from the fact that you computed first the antiderivative
<span class="math-container">$$I(x)=\int \frac{\log \left(1-x^2\right)}{(1-\beta x)^2}\,dx$$</span> which gives
<span class="math-container">$$ I(x)=\frac1\beta\left(\frac{\log \left(1-x^2\right)}{1-\beta x}-\... |
1,809,871 | <p>I am trying to solve Poisson equation using FFT. The issue appears at wavenumber $k = 0$ when I want to get inverse Laplacian which means division by zero.</p>
<p>We have </p>
<p>${\nabla ^2}\phi = f$ </p>
<p>Taking FFT from both side we get:</p>
<p>$-k^2\hat\phi = \hat f $</p>
<p>or </p>
<p>$\hat\phi = \fra... | mathreadler | 213,607 | <p>This is conceptually similar the <strong>integrating constants</strong> that show up when you are solving a differential equation by other methods. Usually these integrating constants are decided by your <strong>boundary conditions</strong>.</p>
<p>Instead of doing any division you can simply rewrite it as a least-... |
1,489,078 | <p>Let $P(S)$ denotes the power set of set $S$. Which of the following is always true$?$</p>
<ol>
<li>$P(P(S)) = P(S)$</li>
<li>$P(S) ∩ P(P(S)) = \{ Ø \}$</li>
<li>$P(S) ∩ S = P(S)$</li>
<li>$S ∉ P(S)$</li>
</ol>
<hr>
<p>I try to explain $:$
If $S$ is the set $\{x, y, z \},$ then the subsets of S are:</p>
<p>${}$ (... | wythagoras | 236,048 | <p>No, <span class="math-container">$\{\}$</span> and <span class="math-container">$\{\{\}\}$</span> are not the same. Think of sets as bags. <span class="math-container">$\{\}$</span> is an empty bag. <span class="math-container">$\{\{\}\}$</span> is a bag with an empty bag within it. Therefore the outermost bag is no... |
1,123,247 | <p>let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent.Find range of $p$ such that the series $\sum |a_n|^p$ is convergent.</p>
<p>My try:</p>
<p>To show the series it is necessary to show that sequence of partial sums of $\sum |a_n|^p$ is bounded.</p>
<p>Consider $S_n=|a_1|^p+|... | guest | 125,454 | <p>If $p\geq2$ then $\sum|a_n|^p$ converges since $$\lim\limits_{n\to\infty}\frac{|a_n|^p}{|a_n|^2}=\lim\limits_{n\to\infty}|a_n|^{p-2}=0$$ where p>2. On the other hand if $0<p<2$ then there exists a sequcence such that $\sum|a_n|^2$ is convergent but $\sum|a_n|^p$ is divergent. Formally, for all $p\in(0,2)$ ther... |
1,405,809 | <p>So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L'Hopital's rule.<br>
I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.</p>
| GeorgSaliba | 142,772 | <p>$$x=\log_a(t)$$</p>
<p>$$\lim_{t\rightarrow 1}\frac{t-1}{\log_at}=\lim_{t\rightarrow 1}\frac{(t-1)\log(a)}{\log(t)}=\log(a)$$
EDIT:</p>
<p>$$L=\lim_{x\rightarrow 1}\frac{\log(x)}{x-1}=\lim_{x\rightarrow 0}\frac{\log(x+1)}{x}=\lim_{x\rightarrow 0}\int_0^x \frac 1x\frac{dt}{1+t}$$
Put: $t=xs$ then:
$$L=\lim_{x\right... |
1,405,809 | <p>So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L'Hopital's rule.<br>
I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.</p>
| Paramanand Singh | 72,031 | <p>In order to evaluate the limit $$\lim_{x \to 0}\frac{a^{x} - 1}{x}\tag{1}$$ for $a > 0$, it is necessary to have a proper definition of $a^{x}$. This has been pointed out in one of the comments to the question, but OP has perhaps not understood the necessity of such a comment. In case you want to evaluate the lim... |
1,405,809 | <p>So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L'Hopital's rule.<br>
I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.</p>
| aman | 264,003 | <p>Convert it into standard limit as write $a^x$ as $e^{x\ln a}$ and
$\lim_{x \to 0} \dfrac{e^{x-1}}{x}=1$</p>
|
2,140,965 | <p>Arithmetic progression
:$a_n=\{a,a+d,a+2d,...,a+nd,...\}$</p>
<p>Geometric progression
:$b_n=\{b,bq,bq^2,...,bq^n,...\}$</p>
<p>if : $a_r,a_s,a_t,$ be a Geometric progression($b_n$)</p>
<p>then : What is q?</p>
<p>my way :
let :$r>s>t$</p>
$$ a_{r} =a +(r-1)d $$
$$ a_{s} =a +(s-1)d $$
$$ a_{t} =a +(t-1... | ajotatxe | 132,456 | <p>Let $\mathbf w'$ be a vector orthogonal to $\mathbf w$, for example $\mathbf w'=(2,-1)$.</p>
<p>Find coefficients $\alpha$ and $\beta$ such that $\mathbf v=\alpha\mathbf w+\beta\mathbf w'$. The vector $\alpha\mathbf w$ is the projection.</p>
<p>The second question looks the same, but note that $\mathbf v$ and $\ma... |
411,549 | <p>Here is an modular equation</p>
<p>$$5x \equiv 6 \bmod 4$$</p>
<p>And I can solve it, $x = 2$.</p>
<p>But what if each side of the above equation times <strong>8</strong>, which looks like this</p>
<p>$$40x \equiv 48 \bmod 4$$</p>
<p>Apparently now, $x = 0$. Why is that? Am I not solving the modular equation in... | Alex Wertheim | 73,817 | <p>Modular equivalence classes are multiplicative. Hence, since $40 \equiv 0 (\text{mod } 4)$ and $48 \equiv 0 (\text{mod } 4)$, all you've written there is $0\cdot x \equiv 0 (\text{mod } 4)$, which is true for any $x \in \mathbb{Z}/4\mathbb{Z}$ (the set of equivalence classes mod $4$). $x = 0$ is not the only "soluti... |
1,818,557 | <p>I will be teaching some "topology" to high school students. I was wondering how to explain to such a school student that on a sphere the shortest path between 2 points is given by a great circle?</p>
<p>Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot fin... | Mikhail Katz | 72,694 | <p>I have found it helpful to replace the sphere by an apple and introduce an "internal" observer by placing an ant on the apple. The ant will crawl from point $A$ to point $B$ on the sphere by following the shortest path (the queen can't wait) which is always an arc of great circle.</p>
<p>An additional point that st... |
4,358,080 | <p>I'm making a pump and I need to make a circular (from the front perspective) hole in a side of a pipe. I can't use a drill and I have to print out a shape that I will stick onto it and cut and file away. Will this circle projected off center onto a cylinder be an ellipse, or is it not an exact ellipse and I have to ... | Jean Marie | 305,862 | <p>A) Let us consider first the case of intersecting axes (even, as Jean-Claude Arbaut has pointed to, they aren't intersecting in the question). In fact, the equation of the unwrapped intersection curve is a general oval curve (see equation (*)) and even more complicated curves with equation (**) which are not ellipse... |
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| Shuchang | 91,982 | <p><strong>HINT:</strong> Square both sides and you obtain $\sin 2x=0$ and then...</p>
|
702,506 | <p>We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.</p>
<p>How to solve</p>
<p>$$\sin x - \cos x = -1$$</p>
<p>for the interval $(0, 2\pi)$.</p>
| MPW | 113,214 | <p>Use the fact that $\sin x = \frac{e^{ix}-e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix}+e^{-ix}}{2}$ to rewrite the equation as (after simplifying)
$$e^{2ix} -(1-i)e^{ix}-(1+i)=0$$
Viewing this as a quadratic equation in $e^{ix}$, use the quadratic formula to get
$$e^{ix}= \frac{(1-i)\pm(1+i)}{2}$$
$$e^{ix} = 1 \text{ or ... |
716,767 | <p>I feel like an idiot for asking this but i can't get my formula to work with negative numbers</p>
<p>assume you want to know the percentage of an increase/decrease between numbers</p>
<pre><code>2.39 1.79 =100-(1.79/2.39*100)=> which is 25.1% decrease
</code></pre>
<p>but how would i change this fo... | amWhy | 9,003 | <p>Perhaps this "formula" will be easier to understand (this formula is equivalent to your formula - each can be derived from the other):</p>
<p>$$\dfrac{\text{original value} \;- \;\text{final value}}{\text{original value}} \times 100\% = \text{percent change}$$</p>
<p>That change will be </p>
<ul>
<li><p>an <em>in... |
2,317,406 | <p>How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square?
Any help would be much appreciated.</p>
| Atul Mishra | 396,163 | <p>See the two numbers whose sum you want</p>
<p>That is $10a+b$ and $10b+a$ </p>
<p>Their sum is $11(a+b)$</p>
<p>Since $11$ is a prime ,, so $(a+b)$ must equal to $11$ for being a perfect square</p>
<p>So the numbers whose sum of digits is $11$ will satisfy your condition</p>
<p>Numbers are $29,92, 38,83,47,74... |
4,107,656 | <p>Let <span class="math-container">$G$</span> be a finite abelian group with a neutral element <span class="math-container">$e$</span>. Prove that for any element <span class="math-container">$g$</span> of <span class="math-container">$G$</span>: <span class="math-container">$g^{|G|}=e$</span>. <span class="math-conta... | Vercingetorix | 848,746 | <p>So first we show that multiplication with <span class="math-container">$g$</span> is a bijection, that is, the map <span class="math-container">$h \mapsto gh$</span> is a bijection. We leverage basic properties of groups here:</p>
<p>First suppose there is <span class="math-container">$h_1, h_2$</span> such that <sp... |
4,473,264 | <p>I have part of a circle described by three two dimensional vectors.</p>
<ul>
<li>start point <code>s1</code></li>
<li>center point <code>c1</code></li>
<li>end point <code>e</code></li>
</ul>
<p>I move the start point <code>s1</code> by <code>m1</code>, which is a <strong>known</strong> two dimensional vector. My qu... | Daniel R. Collins | 266,243 | <p>To answer the question in the title (completely separate from the question of elementary-function antiderivatives):</p>
<blockquote>
<p>Does every function have an integral?</p>
</blockquote>
<p>No, there are functions that do not have any defined integral at all. But the exact category of such functions depends on ... |
2,783,129 | <p>How many different value of x from 0° to 180° for the equation $(2\sin x-1)(\cos x+1) = 0$?</p>
<p>The solution shows that one of these is true:</p>
<p>$\sin x = \frac12$ and thus $x = 30^\circ$ or $120^\circ$ </p>
<p>$\cos x = -1$ and thus $x = 180^\circ$</p>
<p><strong>Question:</strong> Inserting the $\arcsin... | Meow Mix | 193,986 | <p>The function $\arcsin$ returns the angle <em>within the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$</em>. In fact, there are infinitely many angles whose sine is $1/2$. Since a function can't spit out more than one number however, we define the $\arcsin$ function to return the inverse of the sine on that interval.</p... |
1,870,305 | <p>I am trying to calculate the Jacobian of a function that has quaternions and 3D points in it. I refer to quaternions as <span class="math-container">$q$</span> and 3D points as <span class="math-container">$p$</span>
<span class="math-container">$$h_1(q)=A C(q)p $$</span>
<span class="math-container">$$h_2(q)=q_1\ot... | cactus314 | 4,997 | <p>Eq (163) in your <a href="http://www.iri.upc.edu/people/jsola/JoanSola/objectes/notes/kinematics.pdf" rel="nofollow">notes</a> has the really amazing equation:</p>
<p>$$ \frac{\partial(\mathbf{q} \otimes \mathbf{a} \otimes \mathbf{q\ast})}{\partial \mathbf{q}} =
\frac{\partial(\mathbf{R} \mathbf{a})}{\partial \mat... |
1,870,305 | <p>I am trying to calculate the Jacobian of a function that has quaternions and 3D points in it. I refer to quaternions as <span class="math-container">$q$</span> and 3D points as <span class="math-container">$p$</span>
<span class="math-container">$$h_1(q)=A C(q)p $$</span>
<span class="math-container">$$h_2(q)=q_1\ot... | Joan Solà | 229,373 | <p>There are different ways to answer your question, but you probably want one of these two: </p>
<ol>
<li><p>You want the derivative with respect to the 4 components of the quaternion q=w+ix+iy+iz, that is, with respect to a 4 vector <span class="math-container">$v_q=(w,x,y,z)^\top\in R^4$</span>.
We have the deriva... |
1,217,771 | <p>Let $x,y \in R$.
If $0 \leq y < x$ for all $x > 0$, then $y=0$.</p>
<p>Proof by contradiction: </p>
<p>Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$".
Subtract $x$ from each part of the inequality to get,
$0-x \leq y-x < 0$
Th... | user21820 | 21,820 | <p><strong>Useful inequality chain</strong></p>
<p>Given $a \ge 0$ and $b > -\frac{1}{2}a$:</p>
<p>  $a+b-2\frac{b^2}{a} \le a+b-\frac{b^2}{a+b} = \frac{a^2+2ab}{a+b} \le \sqrt{a^2+2ab} \le a+b-\frac{1}{2}\frac{b^2}{a+b} \le a+b$</p>
<p><strong>Solution</strong></p>
<p>Given $x \ge 1$:</p>
<p>  $\sqrt... |
138,866 | <p>I have data in a csv file. The first row has labels, and the first column, too.</p>
<pre><code>Datos = Import["C:\\Users\\jodom\\Desktop\\Data.csv"]
</code></pre>
<p>Tha data in the csv file is that:</p>
<pre><code>{{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87}, {4,
123, 23}, {5, 587, 45}, {6, 4... | jodomofo | 46,969 | <p>I have already do something.... </p>
<pre><code>data1 = {{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87}, {4, 123, 23}, {5, 587, 45}, {6, 435, 89}, {7, 896, 65}, {8, 125, 45}, {9, 476, 27}, {10, 987, 80}}
data2 = Drop[data1, 1]
data3 = Dataset[data2]
data4 = data3[All, <|"No" -> 1, "Vol" -&... |
150,084 | <p>I need to say whether or not $f_n(x)=n\left(\sqrt{x+\frac{1}{n}}-\sqrt{x}\right)$ is uniformly convergent on $(0,\infty)$.</p>
<p>I've found that the function is locally convergent to $f(x)=\frac{1}{2\sqrt{x}}$ and was trying to find $\sup{|f_n(x)-f(x)|}$.</p>
<p>I got the derivative $f_n'(x)= \frac{2nx\left(x-\sq... | Zarrax | 3,035 | <p>By the definition of uniform convergence, if they did uniformly converge to a limit function $f$ there would be some $N$ such that if $n \geq N$ then $|f_n(x) - f(x)| < 1$ for all $x$. In particular this would hold for $n = N$ itself. Since $f_N(x)$ is a bounded function, this means so is $f(x)$. So we can let $M... |
150,084 | <p>I need to say whether or not $f_n(x)=n\left(\sqrt{x+\frac{1}{n}}-\sqrt{x}\right)$ is uniformly convergent on $(0,\infty)$.</p>
<p>I've found that the function is locally convergent to $f(x)=\frac{1}{2\sqrt{x}}$ and was trying to find $\sup{|f_n(x)-f(x)|}$.</p>
<p>I got the derivative $f_n'(x)= \frac{2nx\left(x-\sq... | mrf | 19,440 | <p>As already pointed out by several others, the convergence is not uniform on $(0,\infty)$ but it may be worth noting that it is uniform on every subinterval $[c,\infty)$ for $c > 0$.</p>
<p>This follows from the following computation:</p>
<p>$$
\begin{split}
|f_n(x) - f (x)| &= \left|\frac{1}{\sqrt{x+\dfrac... |
701,176 | <p>A function $f$ is differentiable over its domain and has the following properties:</p>
<ol>
<li><p>$\displaystyle f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$</p></li>
<li><p>$\lim_{h \to 0} f(h) = 0$</p></li>
<li><p>$\lim_{h \to 0} f(h)/h = 1$</p></li>
</ol>
<p>i) Show that $f(0)=0$</p>
<p>ii) show that $f'(x)=1+[f(x)]^... | Lord Soth | 70,323 | <p>Perhaps not the shortest or the greatest argument, but it may be helpful to the OP. Let $f(a) = a^3-6a^2-2$. We take the derivative to obtain $f'(a) = 3a^2-12a$. The roots of the derivative are $0$ and $4$. Looking at the sign of the derivative, we observe that $f$ is increasing on $(-\infty,0)$, decreasing on $[0,4... |
235,319 | <p>The problem states: Suppose $f'(b) = M$ and $M <0$. Find $\delta>0$ so that if $x\in (b-\delta, b)$, then $f(x) > f(b).$</p>
<p>This intuitively makes sense, but I am not exactly sure how to find $\delta$. I greatly appreciate any help I can receive. </p>
| Gerry Myerson | 8,269 | <p>The answer is the coefficient of $x^{20}$ in $$(x+x^2+\cdots+x^7)(1+x+\cdots+x^7)^5$$ Do you see why? Then the first bracket is $x(1-x^7)/(1-x)$, and then second is the 5th power of $(1-x^8)/(1-x)$. OK? So now you need to be able to expand $x(1-x^7)(1-x^8)^5(1-x)^{-6}$, and pick out the coefficient of $x^{20}$. Can ... |
383,735 | <p>Given this (very) tricky determinant, how can we calculate it easily?</p>
<p>$$\begin{pmatrix} \alpha + \beta & \alpha \beta & 0 & ... & ... & 0 \\ 1 & \alpha + \beta & \alpha \beta & 0 & ... & 0 \\ 0 & 1 & \alpha + \beta & \alpha \beta & ... & ... \\ ...... | vadim123 | 73,324 | <p>Let $D_n$ represent the determinant of this $n\times n$ matrix. Expanding on the first column, we see that $D_n=(\alpha+\beta)D_{n-1}-\alpha\beta D_{n-2}$, where the second is found by expanding on the first row of the resulting minor. The recurrence begins with $D_1=\alpha+\beta, D_2=(\alpha+\beta)^2-\alpha\beta=... |
670,570 | <p>Four gallons of yellow paint plus two gallons of red paint make orange paint. I assume this makes six gallons. So the ratio is 4:2, or 2:1.</p>
<p>Question: how many gallons of yellow paint, and how many gallons or red paint, to make two gallons of orange paint?</p>
<pre><code>2y + r = 2
2y + y = 2
3y = 2
y = 2... | André Nicolas | 6,312 | <p>It is not hard to see that $x^5-1$ divides $x^{55}-1$. </p>
<p>Thus $(x-1)(x^4+x^3+x^2+x+1)$ divides $(x^{11}-1)(x^{44}+x^{33}+x^{22}+x^{11}+1)$ and therefore </p>
<p>$x^4+x^3+x^2+x+1$ divides $(x^{10}+x^9+\cdots +x+1)( x^{44}+x^{33}+x^{22}+x^{11}+1)$.</p>
<p>But $x^4+x^3+x^2+x+1$ and $x^{10}+x^9+\cdots +x+1$ are... |
670,570 | <p>Four gallons of yellow paint plus two gallons of red paint make orange paint. I assume this makes six gallons. So the ratio is 4:2, or 2:1.</p>
<p>Question: how many gallons of yellow paint, and how many gallons or red paint, to make two gallons of orange paint?</p>
<pre><code>2y + r = 2
2y + y = 2
3y = 2
y = 2... | Bill Dubuque | 242 | <p>${\rm mod} \dfrac{(x^5\!-\!1)}{(x\!-\!1)}:\,\ \color{#c00}{x^5\equiv 1}\,\Rightarrow\,x^{11n}\equiv x^n (\color{#c00}{x^5})^{2n}\equiv x^n\Rightarrow\,x^{44}\!+\!x^{33}\!+\cdots+1\equiv x^4\!+\!x^3\!+\cdots+1\equiv 0$</p>
|
3,310,812 | <p>Solve the equation :
<span class="math-container">$$p^k=kl+1, $$</span> with <span class="math-container">$p$</span> a prime number and <span class="math-container">$k,l\ge 1$</span> two integers.</p>
<p>I know that <span class="math-container">$(p,k,l)=(3,1,2)$</span> is a solution, but can we find all solutions t... | Henry | 6,460 | <p>Intuitively I feel that the heights and widths should each be in the ratio <span class="math-container">$\sqrt{a}:\sqrt{b}:\sqrt{c}$</span></p>
<p>and so the widths should be <span class="math-container">$\sqrt{a}\frac{W}{\sqrt{a}+\sqrt{b}+\sqrt{c}}$</span>, <span class="math-container">$\sqrt{b}\frac{W}{\sqrt{a}+\... |
738,934 | <p>Write an equation for the line tangent to the graph of x=y^2+4 at the point (5,1).</p>
<p>I have found the derivative y'=1/2y but I do not know what to do next.</p>
| Community | -1 | <p><strong>Hint:</strong> In general, the tangent to the curve $y=f(x)$ at $(x_1,y_1)$ is
$$y=y_1+\left.\dfrac{dy}{dx}\right|_{x=x_1}(x-x_1)=y_1+f^\prime(x_1)(x-x_1)$$</p>
|
738,934 | <p>Write an equation for the line tangent to the graph of x=y^2+4 at the point (5,1).</p>
<p>I have found the derivative y'=1/2y but I do not know what to do next.</p>
| Berci | 41,488 | <p>The curve is $x=y^2+4$, it indeed contains $(5,1)$. Its differential is
$$dx=2y\,dy$$
(Consequently, $\displaystyle\frac{dy}{dx}\ =\ \frac1{2y}\ $ indeed.)</p>
<p>Now, at $y=1$, it gives $dy/dx=1/2$, this is the slope of the curve at that point (note that value of $y$ uniquely determines value of $x$).</p>
<p>Fina... |
1,830,867 | <p>I'm trying to self-study some algebraic topology, reading Hatcher. His questions seem much less straightforwardly worded than Munkres - with Munkres it was always clear that you weren't expected to know much coming in, whereas Hatcher seems to assume a lot of knowledge. Often simple seeming statements of his just se... | Ronnie Brown | 28,586 | <p>The natural environment for "change of base point" is that of groupoid and the <strong>fundamental groupoid</strong> $\pi_1 X$ of a space $X$. In a groupoid $G$, we have "object groups" $G(x)$for each object $x$ of $G$, and if $a: x \to y$ in $G$, then $a$ determines an isomorphism, "conjugation by $a$", $a_\#: G(x... |
555,051 | <p>I am having trouble understanding the following homework question,</p>
<p>Let $H=\{0,\pm 3, \pm6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$.</p>
<p>I know the answer is $H$, $1+H$, and $2+H$ but I am having difficulty understanding why.</p>
<p>Thank you!</p>
| Pedro | 23,350 | <p>Observe that $H=3\Bbb Z$. The fact that $H,1+H,2+H$ are the only cosets is a restatement of the fact any integer $n$ is of the form $3k+b$ with $b=0,1,2$.</p>
|
1,953,628 | <p>0 choices for the 1st person.
17 choices for the 2nd person (must exclude 1st and his/her two neighbours) </p>
<p>For 2 of these choices of 2nd person, there is one shared neighbour, so 15 remaining choices. (e.g. if they are numbered 1 to 20 in a circle, 1st person is #1, 2nd is #3, then people 20,1,2,3,4 are exc... | Community | -1 | <p>Why not again Stars and Bars theorem 1 with the extension of 2 gaps on the bounds. Theorem 1 because we don't want consecutive bars. Then $n=17, k_{max}=3$. </p>
<p>We don't take two on the bounds because we wrap around a circular table !</p>
<ul>
<li>|xxxxxx|xxxxxx|x : $\binom{17-1}{2}$ , two inside, one at the ... |
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