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 |
|---|---|---|---|---|
2,217,928 | <p>I know that in infinite dimensional Hilbert spaces sometimes the best we can do is to find an orthonormal basis in the sense that any element in H can be approximated arbitrarily close in the NORM by a finite linear combination of this basis elements.</p>
<p>So then does that mean we can't expect that every x in H ... | Ben Grossmann | 81,360 | <p>There are two notions of a basis. One is the idea of a Hamel basis, and the other is the idea of a Schauder basis. For more on that, see <a href="https://math.stackexchange.com/questions/630142/what-is-the-difference-between-a-hamel-basis-and-a-schauder-basis">this post</a>. Every vector space has a Hamel basis. ... |
110,709 | <p>By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if they have the same image in $(M,g)$. </p>
<p>Manifolds with constant curvature $\leq 0$,
by Cartan's theorem,
cannot ha... | Igor Rivin | 11,142 | <p>Ellipsoids with almost but not quite equal axes have exactly three simple closed geodesics.</p>
|
3,642,479 | <p>What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?</p>
| Physical Mathematics | 592,278 | <p>The most important property (IMHO) that <span class="math-container">$\mathbb{R}$</span> has that its extensions don't have is that it is a totally ordered field with least upper bound property. You don't have an order (that works nicely with the field) in these extensions.</p>
|
2,271,204 | <p>How do I solve these two equations:
$\frac{x}{x+y} = \frac{1}{2}$ and $\frac{y}{x+y} = \frac{1}{6}$ ?</p>
<p>I tried reducing the first one and end up getting $x = y$, clearly my maths is pretty dusty </p>
| Community | -1 | <p>Turn it to a linear form</p>
<p>$$2x=x+y,\\6y=x+y.$$</p>
<p>But then,</p>
<p>$$x=3y$$ and substituting,</p>
<p>$$6y=3y+y$$ or $y=x=0$, which is not allowed.</p>
|
2,153,128 | <p>I have been given an exercise to convert switch coordinated from cylindrical to rectangular ones. This task is easy but one of them is a strange looking. The point in cylindrical coordinates is $(0,45,10)$. This corresponds to $r=0$. What is this point? Is it not the origin? But why then the angle and z-coordinate. ... | Shashaank | 333,392 | <p>If you just have to choose then it will be 11C2 and 11C3 respectively . Had you asked for arrangement of these then it would have been 11P2 and 11P3 , because this time 3 choses objects can be arranged in 6 ways also. So you would have to multiply 11C3 * 3! Which is 11P3. </p>
<p>You can think of it in this way -</... |
552,384 | <blockquote>
<p>Prove that</p>
<p>$$
\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx = \ln\left(\frac{a}{b}\right)
$$</p>
</blockquote>
<p><strong>My Attempt:</strong></p>
<p>Define the function $I(a,b)$ as</p>
<p>$$
I(a,b) = \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx
$$</p>
<p>Differentiate both side wit... | Count Iblis | 155,436 | <p>$$
\begin{split}
\int_{0}^{\infty}\frac{\exp(-ax) - \exp(-bx)}{x}dx &= \lim_{\epsilon\to 0}\int_{\epsilon}^{\infty}\frac{\exp(-ax) - \exp(-bx)}{x}dx\\
&=\lim_{\epsilon\to 0}\left[\int_{\epsilon}^{\infty}\frac{\exp(-ax)}{x}dx - \int_{\epsilon}^{\infty}\frac{\exp(-bx)}{x}dx\right]\\
&=\lim_{\epsilon\to 0}\... |
1,940,249 | <p>According to Wikipedia, <a href="https://en.wikipedia.org/wiki/First-order_logic#Completeness_and_undecidability" rel="nofollow">first order logic is complete</a>. What is the proof of this?</p>
<p>(Also, in the same paragraph, it says that its undecidable. Couldn't you just enumerate all possible proofs and dispro... | Noah Schweber | 28,111 | <p>I think there's a bit of confusion about the meaning of the theorem. The phrase "First-order logic is complete" means <strong>exactly</strong> "If a sentence $\varphi$ is true in every model of $\Gamma$, then $\Gamma\vdash\varphi$" <em>(so it's saying something about how the</em> semantics <em>and a</em> <a href="ht... |
30,586 | <p>There are many ways to type a pipe. You could use \$|\$ (<span class="math-container">$|$</span>), \$\vert\$ (<span class="math-container">$\vert$</span>), \$\mid\$ (<span class="math-container">$\mid$</span>), or just a plain | (not surrounded by dollar signs). You could also use vmatrix to indicate matrix determin... | Community | -1 | <p>I believe the word "<a href="https://en.wikipedia.org/wiki/Vertical_bar#Pipe" rel="nofollow noreferrer">pipe</a>" is used more in computer science than in mathematics. </p>
<p>Unlike the strict syntax requirement in programming languages, <em>how</em> to type out this vertical bar in mathematical writing is mostly ... |
3,831,310 | <p>I try to integrate</p>
<p><span class="math-container">$$\int \frac {dv}{\frac {-c}{m}v^2 - g \sin \theta}$$</span></p>
<p>I did substituted <span class="math-container">$u = \frac{c}{m}$</span> and <span class="math-container">$w = g \sin \theta$</span> to get</p>
<p><span class="math-container">$$-\int \frac {dv}{... | Soumyadwip Chanda | 823,370 | <p>Divide the numerator by <span class="math-container">$u$</span> to get <span class="math-container">$$-\frac{1}{u}\int_{ }^{ }\frac{dv}{v^{2}+\left(\frac{w}{u}\right)}$$</span></p>
<p>Now you may apply the standard integral, a list of which is given <a href="http://integral-table.com/downloads/integral-table.pdf" re... |
1,016,302 | <p>I'm having a bit of trouble with a the proof in Ross' Elementary Analysis. The theorem is the $\epsilon-\delta$ one. Theorem: Let $f$ be continuous at $x_0$ in $dom(f)$ if and only if for each $\epsilon>o$ there exists $\delta>0$ such that $x \in dom(f)$ and $|x-x_0|<\delta$ implies $|f(x)-f(x_0)|$ $(1)$. <... | 2good4this | 178,838 | <p>His choice of delta is not special in any interesting way, and it's not that particular choice of delta that makes the condition for continuity fail. That delta is just one of the many positive deltas that make the condition fail for a particular epsilon. He assumes that the function is continuous but the condition ... |
2,368,771 | <blockquote>
<p>How to prove the function $$ f(z)=\exp\Big(\frac{z}{1-\cos z}\Big)$$ has an essential singularity at $z=0$ ?</p>
</blockquote>
<p>It's actually hard to express the Laurent series of $f(z)$ around $0$, because the power $\frac{z}{1-\cos z}$ itself is already in the series form (since $\cos z$ appears ... | mathstackuser12 | 361,383 | <p>$\exp \left( \frac{z}{1-\cos \left( z \right)} \right)=\exp \left( \frac{2}{z}+\frac{z}{6}+\frac{{{z}^{2}}}{120}+... \right)=\exp \left( \frac{2}{z}+O\left( z \right) \right)=\exp \left( O\left( z \right) \right)\sum\limits_{n=0}^{\infty }{\frac{{{2}^{n}}}{n!{{z}^{n}}}}$</p>
<p>...or am i missing something?</p>
<p... |
77,642 | <p>$$\frac{1}{\sin(z)} = \cot (z) + \tan (\tfrac{z}{2})$$</p>
<p>I did this: </p>
<p><strong>First attempt</strong>: $$\displaystyle{\frac{1}{\sin (z)} = \frac{\cos (z)}{\sin (z)} + \frac{\sin (\frac{z}{2})}{ \cos (\frac{z}{2})} = \frac{\cos (z) }{\sin (z)} + \frac{2\sin(\frac{z}{4})\cos(\frac{z}{4})}{\cos^{2}(\frac{... | robjohn | 13,854 | <p>Start out with
$$
\frac{1-\cos(z)}{\sin(z)}=\frac{2\sin^2(\tfrac{z}{2})}{2\sin(\tfrac{z}{2})\cos(\tfrac{z}{2})}=\tan(\tfrac{z}{2})\tag{1}
$$
and add $\cot(z)$ to both sides:
$$
\frac{1}{\sin(z)}=\cot(z)+\tan(\tfrac{z}{2})\tag{2}
$$</p>
|
1,782,423 | <p>Gödel's completeness theorem says that for any first order theory $F$, the statements derivable from $F$ are precisely those that hold in all models of $F$. Thus, it is not possible to have a theorem that is "true" (in the sense that it holds in the intersection of all models of $F$) but unprovable in $F$.</p>
<p>H... | Roy Simpson | 40,335 | <p>Although it is the case that the Completeness Theorem of First Order Logic implies that : </p>
<p><em>Every valid sentence $\varphi$ has a proof</em>.</p>
<p>and that such proofs can be found semi-decidably - there <em>is</em> a sense in which the necessity of using weak Konig's Lemma ($WKL_{0}$) in the meta-proof... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| Owen | 125,714 | <p>Singular Value Decomposition, which is one of the most widely-used techniques in numerical computing, is a basis-change. It takes a linear transformation, and does a basis change on both the input space and the output space so that the transformation becomes just multiplication with no addition (ie a diagonal matrix... |
219,782 | <p>Hi I'm wondering if there's some workaround to get Mathematica to use time-shifting identities for Laplace and Inverse Laplace transforms. My examples are below.</p>
<pre><code>$Assumptions = Element[{t, τ}, PositiveReals];
LaplaceTransform[f[t -τ]HeavisideTheta[t-τ], t, s]
(* Output *)
(* LaplaceTransform[f[t -τ]... | Nasser | 70 | <p><a href="https://i.stack.imgur.com/saN4W.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/saN4W.gif" alt="enter image description here"></a></p>
<pre><code>fApprox[max_, t_] := (1/2) +
Sum[ Sin[2 n Pi]/(Pi n) Cos[
n Pi t] + ((-1)^n - Cos[2 Pi n])/(Pi n) Sin[n Pi t], {n, 1, max}]
f[t_] :=... |
2,625,719 | <p>I'm getting an equation $$(\ln y - x)\frac{dy}{dx} - y\ln y = 0$$</p>
<p>Which I try to factorize and bring over to get:
$$\frac{dy}{dx} = \frac{y\ln y}{(\ln y - x)}$$</p>
<p>But this cannot be factorized further. I am intending to use either the the substitution $u=y/x$ or finding the integrating factor, but this... | user577215664 | 475,762 | <p><strong><em>Hint</em></strong></p>
<p>Substitute $y=e^z$
$$(\ln y - x)\frac{dy}{dx} - y\ln y = 0$$
$$(z - x)\frac{dy}{dz} \frac{dz}{dx}- e^zz = 0$$
$$(z - x)e^z z'- e^zz = 0$$
$$(z - x) z'-z = 0$$
$$z'=\frac z {z-x}$$
Then consider $\frac {dx}{dz}$ instead:
$$x'+\frac x z=1$$
Multiply by z $(z \ne 0)$
$$ x'z+x=z$$
... |
1,910,030 | <p>Let $V$ be a vector space and $U,W$ be its subspaces. Prove that if the union $U\cup W$ is a subspace of $V$, then $W \subseteq U$ or $U \subseteq W$.</p>
<p>I'm not sure where to begin at all really.</p>
| Kitter Catter | 166,001 | <p>Broadly speaking the reason we are multiplying is that each of those choices is in some respect independent. Since our choice of ranks doesn't affect the number of cards they are independent. </p>
<p>As to the 44 issue I think others have covered that neatly, since we only want two pair and not a full house we can'... |
2,614,375 | <p>I have tried using prime factorization which is $2^2 \times 5^2 \times 11$ and found out that I'll need $37$ slices. but in the question paper which is a multiple choice question there's no such answer. the choices are $30,31,32,61,110$.</p>
| Community | -1 | <p>You are asked to count the slices, not the cuts.</p>
<p>Hence with $10\times10\times11=1100$,$$10+10+11=31.$$</p>
|
424,853 | <p>As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been a narrowing of focus, via a choice to ignore certain "pathological"—or at least intractably complicated—ph... | Dmitri Pavlov | 402 | <p>Such a timeline is necessarily highly subjective.</p>
<p>With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory.
The list below concentrates on developments that in some way affect the foundations of homotopy theory, as opposed to general advances in h... |
613,836 | <blockquote>
<p>Let $x,y,z$ be integers and $11$ divides $7x+2y-5z$. Show that $11$
divides $3x-7y+12z$.</p>
</blockquote>
<p>I know a method to solve this problem which is to write into $A(7x+2y-5z)+11(B)=C(3x-7y+12z)$, where A is any integer, B is any integer expression, and C is any integer coprime with $11$.</... | mathlove | 78,967 | <p>Let $n=3x-7y+12z$. Since there exsits an $m\in\mathbb Z$ such that
$$7x+2y-5z=11m,$$
We have the following two :</p>
<p>$$7n=21x-49y+84z$$
$$33m=21x+6y-15z$$
Then, we have $$7n-33m=-55y+99z\Rightarrow 7n=33m-55y+99z=11(3m-5y+9z).$$
Since we know $7$ and $11$ are coprime, we know that $n$ is a multiple of $11$.</p... |
4,045,943 | <p>I was requested to find the values of the parameter p for which the following series converges:</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \sqrt{n} \ln^{p} \left(1+ \frac{1}{\sqrt{n}}\right)$$</span></p>
<p>I tried using <a href="https://en.wikipedia.org/wiki/Root_test" rel="nofollow noreferrer">Cauchy'... | VIVID | 752,069 | <p><span class="math-container">$$\begin{align}
\sqrt{n} \ln^{p} (1+ \frac{1}{\sqrt{n}})
&\overset1= \sqrt n \left( \frac{1}{\sqrt n} + O\left(\frac 1n \right)\right)^p \\
&\overset2= \sqrt n\frac{1}{(\sqrt n)^p}\left(1 + O\left(\frac{1}{\sqrt n}\right)\right) \\
&\overset3= \frac{1}{n^{p/2-1}} + O\left(\f... |
4,045,943 | <p>I was requested to find the values of the parameter p for which the following series converges:</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \sqrt{n} \ln^{p} \left(1+ \frac{1}{\sqrt{n}}\right)$$</span></p>
<p>I tried using <a href="https://en.wikipedia.org/wiki/Root_test" rel="nofollow noreferrer">Cauchy'... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$\frac{\sqrt{n} \ln^{p} \left(1+ \frac{1}{\sqrt{n}}\right)}{\left(\frac1{\sqrt{n}}\right)^{p-1}}=\left(\frac{\ln \left(1+ \frac{1}{\sqrt{n}}\right)}{\frac1{\sqrt{n}} }\right)^p \to 1$$</span></p>
<p>therefore by limit comparison test the given series converges if and... |
3,274,061 | <p>If <span class="math-container">$T:H \to H$</span> is linear and bounded and compact and <span class="math-container">$\{h_n\}$</span> is bounded, is <span class="math-container">$\{T(h_n)\}$</span> a compact subset of <span class="math-container">$H$</span>? We have <span class="math-container">$H$</span> as a Hilb... | Nate Eldredge | 822 | <p>Not necessarily. Take <span class="math-container">$H = \ell^2$</span> and let <span class="math-container">$(Th)(i) = \frac{1}{i} h(i)$</span>. You can verify that <span class="math-container">$T$</span> is compact. Let <span class="math-container">$\{h_n\}$</span> be the usual orthonormal basis for <span class=... |
3,274,061 | <p>If <span class="math-container">$T:H \to H$</span> is linear and bounded and compact and <span class="math-container">$\{h_n\}$</span> is bounded, is <span class="math-container">$\{T(h_n)\}$</span> a compact subset of <span class="math-container">$H$</span>? We have <span class="math-container">$H$</span> as a Hilb... | Andreas Blass | 48,510 | <p>Remember that a finite-dimensional vector space with an inner product is a perfectly good Hilbert space, and all its linear operators are compact. So you can get a counterexample by taking <span class="math-container">$H$</span> to be <span class="math-container">$\mathbb R$</span> or <span class="math-container">$\... |
269,696 | <p>I am given that $\sum\limits_{n=1}^\infty a_n$ is convergent. </p>
<p>I need to determine whether $\sum\limits_{n=1}^\infty (a_n)^\frac{1}{3}\;$ and $\;\sum\limits_{n=1}^\infty (a_n)^2\;$ are also convergent.</p>
<p>Imagine that $a_n = \dfrac{1}{n^4}.\;$ I believe that this is convergent because it's converging to... | Sugata Adhya | 36,242 | <p>$\sum\limits_{n=1}^\infty\frac{1}{n^3}$ & $\sum\limits_{n=1}^\infty(-1)^n\frac{1}{\sqrt n}$ are convergent but $\sum\limits_{n=1}^\infty\frac{1}{n}$ is not.</p>
|
1,080,746 | <p>I want to calculate the limit of this sum :</p>
<p>$$\lim\limits_{x \to 1} {\left(x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb\right)}$$</p>
<p>My efforts to solve the problem are described in the <a href="https://math.stackexchange.com/a/1308281/202081">self-answer below</a>.</p>
| achille hui | 59,379 | <p>Based on a paper "Summability of alternating gap series" by J.P. Keating and J.B.Reade in year $2000$,
(an online copy can be found <a href="http://journals.cambridge.org/article_S001309150002071X">here</a>)
, one can use <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula">Poisson summation formula</a> ... |
1,080,746 | <p>I want to calculate the limit of this sum :</p>
<p>$$\lim\limits_{x \to 1} {\left(x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb\right)}$$</p>
<p>My efforts to solve the problem are described in the <a href="https://math.stackexchange.com/a/1308281/202081">self-answer below</a>.</p>
| Abdou Abdou | 202,081 | <ul>
<li>After consulting @achille hui's answer, looking on this <a href="http://journals.cambridge.org/download.php?file=%2FPEM%2FPEM2_43_01%2FS001309150002071Xa.pdf&code=59a6cb2e1c51d86990aecae20ceafeb1" rel="nofollow noreferrer">paper</a> he/she brought, I found an approbation to my answer posted awhile ago at ... |
4,586,314 | <p><em>A three-sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region, as shown in the diagram below. The enclosed area is to equal <span class="math-container">$1800 m^2$</span> and the fence running parallel to the river must be set back at least <span ... | nasekatnasushi | 1,123,549 | <p>The way I understand it is that you want one matrix <span class="math-container">$ T $</span> that does the same kind of thing (as you described) to any matrix <span class="math-container">$ A $</span>.</p>
<p>There is no such matrix <span class="math-container">$ T $</span>. One way you can view multiplying a matri... |
1,649,997 | <p>I'm having a hard time coming up with two unbounded sequences where their difference yields $0$ when $n\rightarrow\infty$. Any ideas?</p>
| P Vanchinathan | 28,915 | <p>Take any monotonic sequence diverging to $+\infty$, as the first sequence. Define the second sequence by adding $\frac1n$ to the $n$th term of the first.
What can you say bout the difference between these two sequences? </p>
|
1,191,176 | <p>I know how to prove the result for $n=2$ by contradiction, but does anyone know a proof for general integers $n$ ?</p>
<p>Thank you for your answers.</p>
<p>Marcus</p>
| Viktor Vaughn | 22,912 | <p>Apply <a href="http://en.wikipedia.org/wiki/Eisenstein%27s_criterion">Eisenstein's criterion</a> with $p = 2$ to the polynomial $f(x) = x^n - 2$. This shows that $f$ is irreducible over $\mathbb{Q}$, so in particular it has no roots in $\mathbb{Q}$.</p>
|
4,383,098 | <p>This was a question I had ever since I started studying Formal mathematics. Take ZFC for example, in it the axioms tell us 'tests' to check if something is a set or not and how the object, if they are set, behave with some other operations defined on the set.</p>
<p>My question is how exactly do we find objects do f... | Graham Kemp | 135,106 | <p>To be demonstrated: <span class="math-container">$(A∪B)∩(B∪C)∩(C∪A)=(A∩B)∪(A∩C)∪(B∩C)$</span></p>
<p>Observation: Elements of the LHS that are in <span class="math-container">$A$</span> must also be in <span class="math-container">$B$</span> or in <span class="math-container">$C$</span>, however, those that are not ... |
4,435,032 | <p>I have equations of two conic sections in general form. Is it possible to find minimal distance between them (if they are not intercross)?</p>
<p>I need it to calculate is two spacecrafts on two orbits (2d case) can collide or not. If minimal distance bigger than sum of radiuses of bounding circles I don't need to c... | Overflowian | 211,748 | <p><strong>Remark1.</strong> Recall that the holonomy of a metric is the holonomy of its Levi-Civita connection. Thus a manifold may admit an affine connection with holonomy in <span class="math-container">$G<GL(n)$</span> and still not admitting any metric with holonomy in <span class="math-container">$G$</span>.</... |
2,634,282 | <p><strong>Question:</strong></p>
<p>Write all the points where $\cfrac{n^2-9n+20}{n^4-n^3-3n^2+n+2}$ is discontinuous.Provide the answer in simplest form.</p>
<p><strong>My Approach:</strong></p>
<p>Should I convert the numerator and denominator into its respective product form and try to cancel them? I am not gett... | David C. Ullrich | 248,223 | <p>Answer to the question below. First we should note that the theorem as stated is false, if we're talking about Riemann integrals.</p>
<p>Let $X=Y=[0,1]$. Let $f(x,y)=0$ if $y\ne0$; let $f(x,0)=1$ if $x$ is rational, $0$ otherwise. It's easy to see from the definition that $f$ is Riemann integrable on $X\times Y$ (o... |
2,634,282 | <p><strong>Question:</strong></p>
<p>Write all the points where $\cfrac{n^2-9n+20}{n^4-n^3-3n^2+n+2}$ is discontinuous.Provide the answer in simplest form.</p>
<p><strong>My Approach:</strong></p>
<p>Should I convert the numerator and denominator into its respective product form and try to cancel them? I am not gett... | RRL | 148,510 | <p>The statement of the Theorem given as a reference is false. As such, it is just a distraction with respect to the Question, which David C. Ullrich has answered by providing a nice example. </p>
<p>I just will focus on the theorem to help clarify beyond what has already been discussed in the comments.</p>
<p>The hy... |
3,059,617 | <p>Note: I apologize in advance for not using proper notation on some of these values, but this is literally my first post on this site and I do not know how to display these values correctly.</p>
<p>I recently was looking up facts about different cardinalities of infinity for a book idea, when I found a post made sev... | David C. Ullrich | 248,223 | <p>The reason it should really be <span class="math-container">$\aleph_\omega$</span> instead of <span class="math-container">$\aleph_{\aleph_0}$</span> is that we think of <span class="math-container">$\aleph_\alpha$</span> for <em>ordinals</em> <span class="math-container">$\alpha$</span>. Yes, <span class="math-cont... |
3,059,617 | <p>Note: I apologize in advance for not using proper notation on some of these values, but this is literally my first post on this site and I do not know how to display these values correctly.</p>
<p>I recently was looking up facts about different cardinalities of infinity for a book idea, when I found a post made sev... | Asaf Karagila | 622 | <p>Ordinals are not cardinals.</p>
<p>Recall Hilbert's hotel. Where you have infinitely many rooms, one for each natural number, and they are all full. And there's a party, with all the guests invited. At some point, after so many drinks, people need to use the restroom.</p>
<p>So someone goes in, and immediately aft... |
2,044,451 | <p>I'm trying to prove the following inequality:
$$- \ln (x) \leq (x)^{-\frac{1}{e}} $$</p>
<p>over $[0, 1]$. I'm not sure how to move forward. I know that the equality is at $x = e^{-e}$. Any help is greatly appreciated.</p>
| Jack D'Aurizio | 44,121 | <p>If we set $x=e^{-t}$, we just have to show that
$$ \forall t\geq 0,\qquad t\leq \exp\frac{t}{e} \tag{1}$$
but that is trivial by convexity: $f(t)=\exp\frac{t}{e}$ is a convex function on $\mathbb{R}$, and the equation of the tangent line at $x=e$ is exactly $g(t)=t$.</p>
|
1,670,816 | <p>Given <span class="math-container">$a, b \in \Bbb R$</span>, consider the following large tridiagonal matrix</p>
<p><span class="math-container">$$M := \begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
0 & b & (a+2)^2 &am... | copper.hat | 27,978 | <p>If you want an approximation, the implicit function theorem is a useful tool.</p>
<p>Let $\phi(x,\epsilon) = (x-1)(x-2)(x-3)(x-4)-\epsilon x^6$. It is not too
difficult to compute
${\partial \phi(4,0) \over \partial x} = 6$, ${\partial \phi(4,0) \over \partial \epsilon} = -4^6$. Hence there is a function $\xi$ def... |
4,092,182 | <p>Let <span class="math-container">$I$</span> be a interval and <span class="math-container">$f:I \rightarrow \mathbb{R}$</span> monotone and surjective prove that <span class="math-container">$f$</span> is continuous.</p>
<p>I tried using the definition of <span class="math-container">$\epsilon$</span>-<span class="m... | user126154 | 126,154 | <p>Given a tensor <span class="math-container">$T$</span> of type <span class="math-container">$(0,q)$</span> and given a vector field <span class="math-container">$X$</span>, the covariant derivative <span class="math-container">$\nabla_XT$</span> is still a tensor of type <span class="math-container">$(0,q)$</span>, ... |
385,404 | <p>So, I have this question which is still troubling me:</p>
<blockquote>
<p>Find the value of $k$ such that the equation $2x^3 + 3x^2 + kx - 48 = 0$ has two solutions equal in value but opposite in sign.</p>
</blockquote>
<p>I've had numerous attempts at this, such as using simultaneous equations and the factor th... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>So, we can assume the roots to be of the form $a,-a,b$</p>
<p>So using <a href="http://mathworld.wolfram.com/VietasFormulas.html">Vieta's formula</a> </p>
<p>$a+(-a)+b=-\frac32\implies b=-\frac32$ and $a\cdot(-a)\cdot b=\frac{48}2\implies a^2=16\implies a=\pm4$</p>
<p>So, the roots are $\pm4,-\frac3... |
1,915,257 | <p>$$\sum_{n=2}^\infty\frac{\cos\ln\ln n}{\ln n}$$
My idea is
$$-\frac1{\ln n}\le\frac{\cos\ln\ln n}{\ln n}\le\frac1{\ln n}$$
But I don't know if $\sum\frac1{\ln n}$ converges.</p>
| marty cohen | 13,079 | <p>Off the top of my head and from my phone:</p>
<p>Cos(log log n)
Is essentially constant for longer and longer stretches
and sum 1/log n diverges so the sum diverges.</p>
<p>I'm sure this could be made rigorous for any function thst grows slowly like log log.</p>
|
4,136,532 | <p>I know it's a weird question.<br />
But this thing is confusing me.<br />
(x̅) : average μ<br />
</p>
<p>①<br />
∵ <span class="math-container">$\frac{1}{n}\sum\limits_{i=1}^n(x_i) = \bar{x}$</span><br />
∴ <span class="math-container">$\sum\limits_{i=1}^n(x_i) = {n}\bar{x}$</span><br />
</p>
<p>②<br />
∵ <s... | user170231 | 170,231 | <p><span class="math-container">$\bar x$</span> is the average, so it's just some number that you can factor out of the summand:</p>
<p><span class="math-container">$$\sum_{i=1}^n\bar x=\bar x\sum_{i=1}^n1=\bar x\left(\underbrace{1+1+\cdots+1}_{n\text{ times}}\right)=\bar x n$$</span></p>
|
1,621,275 | <blockquote>
<p>Show that $a_n=\frac{n+1}{2n}a_{n-1}+1$ given that:</p>
<p>$a_n=1/{{n}\choose{0}}+1/{{n}\choose{1}}+...+1/{{n}\choose{n}}$</p>
</blockquote>
<p>The hint says to consider when $n$ is even and odd. When $n=2k$ I get:</p>
<p>$$a_{n}=1/{{2k}\choose{0}}+1/{{2k}\choose{1}}+...+1/{{2k}\choose{2k}}$$
$... | Community | -1 | <p>Let $$f(x) =\sum_{n=1}^{\infty} a_n x^n =\sum_{n=1}^{\infty} \frac{n+1}{2n} a_{n-1} x^n +\sum_{n=1}^{\infty} x^n =1+\sum_{n=1}^{\infty}\frac{n+2}{2(n+1)} a_n x^{n+1} +\frac{x}{1-x}=\frac{1}{1-x} +\frac{1}{2} \int\sum_{n=1}^{\infty}(n+2) a_n x^n dx =\frac{1}{1-x} +\frac{1}{2}\int \frac{1}{x} \left(\sum_{n=1}^{\infty... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Piyush Grover | 30,684 | <p><a href="http://rads.stackoverflow.com/amzn/click/0821819984"><em>Geometry and the Imagination</em></a> by Hilbert and Cohn-Vossen.</p>
|
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Zurab Silagadze | 32,389 | <p><a href="http://rads.stackoverflow.com/amzn/click/0195105192">What Is Mathematics? An Elementary Approach to Ideas and Methods </a> by Richard Courant and Herbert Robbins</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0821843672">Lessons in Geometry </a> by Jacques Hadamard, and its companion books:
<a hr... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Arash | 39,212 | <p>I personally enjoyed these books:</p>
<p><a href="http://en.wikipedia.org/wiki/How_to_Solve_It" rel="nofollow">How To Solve It</a> by George Polya</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0883856190" rel="nofollow">Geometry Revisited</a> by H. S. M. Coxeter , Samuel L. Greitzer </p>
|
2,378,042 | <p>I have been learning that the sum of the squares of the first <span class="math-container">$n $</span> natural numbers is given by <span class="math-container">$$\Sigma= \frac {1}{6} n (n+1)(2n+1) $$</span> while the sum of the cubes of the first <span class="math-container">$n $</span> natural numbers is given by:... | Thiago Nascimento | 226,141 | <p>You can use: $\sum_{k=0}^{n} k^{3} = \sum_{k=0}^{n} (k+1)^{3}- (n+1)^{3}$, expand the sum and see that o term cubic cancel and you have a sum in function of $\sum k$. You can use this metod for sum of $k^{4}$ ... etc.</p>
|
28,395 | <p>I want to generate a table of Euclidean distances from the points in a list to a given point, e.g.:</p>
<pre><code>Table[EuclideanDistance[MyList[[i]], c], {i, 1, Length[MyList]}]
</code></pre>
<p>Moreover, when <code>EuclideanDistance</code> returns a value greater than $T$, I'd like to replace that value with a ... | Jens | 245 | <p>Why not use <code>Clip</code>?</p>
<pre><code>MyList = RandomReal[{-1, 1}, 10];
c = {0, 0};
t = 1;
r = -1;
Clip[
Table[EuclideanDistance[MyList[[i]], c], {i, 1, Length[MyList]}],
{0, 1}, {0, r}]
(* ==> {0.9957995322104205`,0.3452581732209688`,0.016464628727136405`,
-1,-1,0.5914902487316964`,0.853121659385... |
429,380 | <p>I am not familiar with this three lines equal sign and reading about it didnt really help with the original problem, which is: </p>
<blockquote>
<p>From the options below choose up to two that show correct solutions
for each of the three calculations:</p>
</blockquote>
<p>The photo of the options:</p>
<blockq... | Higgs88 | 33,088 | <p>I think that I find a proof:</p>
<p>As I have showed, by Jensen's inequality we prove that $\{X_n^+\}_n$ is a backward submartingale. </p>
<p>For any $K>0$, we have</p>
<p>$$KP(|X_n|\geq K)\leq E[|X_n|]=2E[X_n^+]-E[X_n]\leq 2E[X_1^+]-l<\infty$$</p>
<p>It follows that </p>
<p>$$\sup_nP(|X_n|\geq K)\rightar... |
429,380 | <p>I am not familiar with this three lines equal sign and reading about it didnt really help with the original problem, which is: </p>
<blockquote>
<p>From the options below choose up to two that show correct solutions
for each of the three calculations:</p>
</blockquote>
<p>The photo of the options:</p>
<blockq... | Conrado Costa | 226,425 | <p>in the last step one can't choose $K$ large so that
$\sup_{n \geq N} \mathbb{E}[1_{\{X_n^+ \leq -K\}|X_N|}] \leq \frac{\epsilon}{2}$</p>
<p>and that is not just because of a typo (most likely you meant $\sup_{n \geq N} \mathbb{E}[1_{\{X_n \leq -K\}|X_N|}] \leq \frac{\epsilon}{2}$ ) </p>
<p>It is because the negati... |
2,958,566 | <p>Expressing <span class="math-container">$\frac{t+2}{t^3+3}$</span> in the form <span class="math-container">$a_o+a_1t+...+a_4t^4$</span>, where <span class="math-container">$t$</span> is a root of <span class="math-container">$x^5+2x+2$</span>.</p>
<p>So i can deal with the numerator, but how do I get rid of the de... | rtybase | 22,583 | <p><strong>Without induction</strong>. Let's do some tagging 1st
<span class="math-container">$$m \mid a+d \tag{1}$$</span>
<span class="math-container">$$m \mid (b-1)c \tag{2}$$</span>
<span class="math-container">$$m \mid ab-a+c \tag{3}$$</span>
then
<span class="math-container">$$m \mid ab^n+cn+d \iff m \mid a\left(... |
2,958,566 | <p>Expressing <span class="math-container">$\frac{t+2}{t^3+3}$</span> in the form <span class="math-container">$a_o+a_1t+...+a_4t^4$</span>, where <span class="math-container">$t$</span> is a root of <span class="math-container">$x^5+2x+2$</span>.</p>
<p>So i can deal with the numerator, but how do I get rid of the de... | Bill Dubuque | 242 | <p><span class="math-container">$\bmod m\!:\ \color{#0a0}{f_{\large n+1}\!-b f_{\large n}} =\, \overbrace{(1\!-\!b)c}^{\large \equiv\ 0}n \,\overbrace{-\color{#c00}d\,b+\color{#c00}d+c}^{\large \equiv\ 0\ \ {\rm by}\ \ \color{#c00}{d}\ \equiv\ -a}\!\equiv\color{#0a0} 0,\ $</span> so <span class="math-container">$\ f_... |
696,145 | <blockquote>
<p>For each positive integer <span class="math-container">$n$</span>, define <span class="math-container">$f(n)$</span> such that <span class="math-container">$f(n+1) > f(n)$</span> and <span class="math-container">$f(f(n))=3n$</span>. What is the value of <span class="math-container">$f(10)?$</span><... | Anas A. Ibrahim | 650,028 | <p>Well, note that
<span class="math-container">$$f(f(n))=3n \implies f(f(f(n)))=f(3n)=3f(n)$$</span>
So <span class="math-container">$f(1)=2 \implies f(3)=6=f(f(2)) \implies f(2)=3$</span> (because <span class="math-container">$f$</span> is injective, easily proved from <span class="math-container">$f(f(n))=3n$</span>... |
2,395,611 | <p>This is probably an easy question, hanging on some minor detail, but i cannot find a proof for it. I try working through T.A. Springer's book "Linear algebraic groups" and i got stuck at remark 7.1.4. (I know there is already a question asked about the same remark, but concerning a different statement made in it). S... | Simon Weinzierl | 176,964 | <p>$\require{AMScd}$
So i found an answer to this question in Humphrey's "Linear algebraic groups" (around paragraph §25). There a more general proposition is shown, namely:<br>
For an epimorphism $\varphi : G\to G'$, such that $\varphi(T) =: T'$ is the maximal torus of $G'$, and that $\ker(\varphi)$ is contained in ev... |
1,025,079 | <p>Consider two related problems:</p>
<ol>
<li>You have $n$ cannisters that must go into $m$ trucks that can each carry $k$ cannisters. You require that no truck becomes overloaded, and for each cannister, there is a specified subset of trucks in which it may be safely carried. Is there a way to load all $n$ cannister... | Jamie Lannister | 234,745 | <p>Yes, there is a clever trick for approximating solutions to the knapsack problem. Since the knapsack problem is pseudo polynomial, we can find a full approximation scheme for it such that out approximation can be made arbitrarily close. The details are here:</p>
<p><a href="http://math.mit.edu/~goemans/18434S06/kna... |
176,386 | <p>I stumbled across this "dictionary for noncommutative topology" <a href="http://planetmath.org/noncommutativetopology" rel="nofollow">http://planetmath.org/noncommutativetopology</a>
and I would be very interested in learning more on the subject, particularly I'd like to see why these results are true. </p>
<p>Wha... | Nik Weaver | 23,141 | <p>Dixmier's book is a fine, if dated, introduction to C*-algebra, but really not the right place to learn about noncommutative topology. The place you want to go is Alain Connes' <a href="http://www.alainconnes.org/docs/book94bigpdf.pdf" rel="nofollow">Noncommutative Geometry</a>.</p>
<p>Edit: in response to Yemon's ... |
4,335,896 | <blockquote>
<p><span class="math-container">$$\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\frac{1\cdot3\cdot5\cdot7}{2\cdot4\cdot6\cdot8\cdot10}+\cdots$$</span>
is equal to?</p>
</blockquote>
<p><strong>My approach:</strong></p>
<p>We can see that the <span class="math-con... | Eldar Sultanow | 993,738 | <p>I tried to rewrite the sum:</p>
<p><span class="math-container">$$\sum _{n=1}^k \frac{\left(\frac{1}{4}\right)^n (2 n)!}{n! (n+1)!}$$</span></p>
<p>computationally using the Mathematica Code:</p>
<pre><code>Sum[Factorial[2*n]*(1/4)^n/(Factorial[n]*Factorial[n + 1]), {n, 1, k}]
</code></pre>
<p>As a result I got:</p>... |
3,064,943 | <p>Consider the senquence of iid r.v. <span class="math-container">$(Y_k)_{k\geq1}$</span> such that <span class="math-container">$\mathbb{P}(Y_k=1)=\mathbb{P}(Y_k=-1)=\frac{1}{2}$</span> and then consider the process <span class="math-container">$X=(X_k)_{n\geq1}$</span> such that <span class="math-container">$X_n=\su... | Davide Giraudo | 9,849 | <p>One can see that
<span class="math-container">$$
\mathbb E\left[X_n^2\right]=\sum_{k=1}^n\frac 1{k^2}\mathbb E\left[Y_k^2\right]
$$</span>
hence that the martingale <span class="math-container">$\left(X_n\right)_{n\geqslant 1}$</span> is bounded in <span class="math-container">$\mathbb L^2$</span>.</p>
|
2,648,273 | <p>I am reading "Linear Algebra" by Takeshi SAITO.</p>
<p>Why $n \geq 0$ instead of $n \geq 1$?<br>
Why $K^0 = \{0\}$?<br>
Is $K^0 = \{0\}$ a definition or not?</p>
<p>He wrote as follows in his book:</p>
<p>Let $K$ be a field, and $n \geq 0$ be a natural number. </p>
<p>$$K^n = \left\{\begin{pmatrix}
... | Aloizio Macedo | 59,234 | <p>Another way to see as why this definition is natural is that $K^n$ can be seen as the set of functions from a set with $n$ elements to $K$ (and the obvious "pointwise" addition, multiplication by scalar etc).</p>
<p>This way, $K^0$ must be the set of functions from a set of zero elements (empty set) to $K$. This is... |
3,186,192 | <p>Is the function below always positive for <span class="math-container">$0< x <1$</span>? (I am determining if the function requires the modulus sign or not.)</p>
<p><span class="math-container">$$\frac{1}{2}\log\left|\frac{1+\log(x)}{1-\log(x)}\right|$$</span></p>
<p>My first instinct is that it cannot be, b... | Dasherman | 177,453 | <p>This function is actually negative in your interval (plot it to verify).</p>
<p>Note that excluding <span class="math-container">$x=1/e$</span> (for which your fraction does not exist), we have in the interval <span class="math-container">$0<x<1$</span> that <span class="math-container">$log (x)<0$</span>,... |
3,855,521 | <p>Suppose <span class="math-container">$f:A\bigoplus M \to A\bigoplus N$</span> and <span class="math-container">$g: A \to A$</span>, where <span class="math-container">$g = \pi_A\circ f\circ l_A$</span>, are isomorphisms of <span class="math-container">$R$</span>-module. Prove that <span class="math-container">$M\con... | Dave L. Renfro | 13,130 | <p>First, let's determine the possible values for <span class="math-container">$x.$</span> Thanks to @Intelligenti pauca for pointing out this oversight in my original answer, which caused significant <em>qualitative</em> errors in my original answer.</p>
<p>Since <span class="math-container">$y^2$</span> is non-negati... |
606,380 | <blockquote>
<p>If <span class="math-container">$abc=1$</span> and <span class="math-container">$a,b,c$</span> are positive real numbers, prove that <span class="math-container">$${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1\,.$$</span></p>
</blockquote>
<p>The whole problem is in the title. If you wanna h... | math110 | 58,742 | <p>I have other nice Cauchy-Schwarz inequality solve it.</p>
<p>since
$$\dfrac{1}{1+a+b}=1-\dfrac{a+b}{1+a+b}$$
so the original inequality can be written
$$\sum_{cyc}\dfrac{a+b}{a+b+1}\ge2$$
use Cauchy-Schwarz inequaliy and the AM-GM inequality,we have
$$\sum_{cyc}\dfrac{a+b}{a+b+1}\ge\dfrac{(\sum\sqrt{a+b})^2}{\sum(a... |
606,380 | <blockquote>
<p>If <span class="math-container">$abc=1$</span> and <span class="math-container">$a,b,c$</span> are positive real numbers, prove that <span class="math-container">$${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1\,.$$</span></p>
</blockquote>
<p>The whole problem is in the title. If you wanna h... | SB1729 | 466,737 | <p>This answer only assumes that <span class="math-container">$abc\geq 1$</span>. Make the following substitution <span class="math-container">$$\sqrt[3]{a}=x,\sqrt[3]{b}=y,\sqrt[3]{c}=z$$</span> then we have <span class="math-container">$xyz\geq1$</span> and we have to prove the following inequality now</p>
<p><span ... |
52,361 | <p>I have seen the following one. Please give the proof of the observation.
We know that, The difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered digits (2nd, 4th...) is divisible by 11. I have checked the same for other numbers in different base system. For example... | Ilmari Karonen | 9,602 | <p>Your question is rather confusingly written, but I assume you're asking for a proof that the base-10 divisibility rule for 11 — sum the even-numbered digits, subtract the odd-numbered digits, check (recursively) if result is divisible by 11 — can be generalized to a divisibility rule for $n+1$ in base $n... |
239,045 | <p>I'm trying to write a short program that does the following over and over again.</p>
<ol>
<li>Import all files in a folder</li>
<li>Automatically generate names for the imported files based on source file name.</li>
<li>Perform a simple operation on the file.</li>
<li>Re-Export the modified files with a new name.</l... | Albert Retey | 169 | <p>This is a very common task for me and most probably many other Mathematica users as well. It typically looks somewhat like the following when I do this:</p>
<pre><code>Scan[
Function[filename,
Module[{data,newdata,newfilename},
Print["working on "<>FileBaseName[filename]<>"...&... |
2,048,962 | <blockquote>
<p><strong>Question:</strong> How would you solve this sinusoidal equation:</p>
<blockquote>
<p>Solve $5\cos(6x)+6=9$. Assume $n$ is an integer and the answers are in degrees.</p>
<ul>
<li><p>$-8.86+n\cdot 60$</p></li>
<li><p>$-3.54+n\cdot 60$</p></li>
<li><p>$3.54+n\cdot 60... | DanielWainfleet | 254,665 | <p><span class="math-container">$\cos 6x=3/5\iff |\cos 3x|=(\sqrt {1+\cos 6x})/2=(\sqrt {8/5})/2=\sqrt {2/5}.$</span></p>
<p><span class="math-container">$|\cos 3x|=\sqrt {2/5}\iff |4\cos^3x-3\cos x|=\sqrt {2/5}\iff (4\cos^3x-3\cos x)^2=4/25.$</span></p>
<p>Let <span class="math-container">$y=\cos^2 x.$</span> Then <s... |
3,823,048 | <p>My questions needs more context than what can fit into the title so let me elaborate. In pretty much all the art textbooks I am reading on linear perspective they state that the correct way of placing an ellipse is to imagine the minor axis of an ellipse as an axel of a wheel running to the opposite vanishing point.... | Jean Marie | 305,862 | <p>Minor/major axes, foci, etc. are "metric objects" that have a meaning in the framework of metric geometry, therefore are outside the scope of projective geometry which (in particular) doesn't preserve neither the lengths nor the ratio of lengths (but the cross-ratio). As a consequence, it is not astonishin... |
2,737,830 | <p>For some fixed b, r > 0, place b blue balls and r red balls in a box. Select balls from the box, one after the other, without replacement. Prove that the probability that the kth ball taken from the box is blue is the same for 1 ≤ k ≤ b.</p>
<p>Does this just come from the fact that if you do not know what you are ... | Christian Blatter | 1,303 | <p>For all $k\in [b+r]$ the probability that the $k^{\rm th}$ ball is blue is the same, namely ${b\over b+r}$.</p>
<p>Imagine that the blue balls are secretly numbered from $1$ to $b$ and the red balls from $1$ to $r$. The devil arranges the $b+r$ now distinguishable balls in a random linear order in one of $(b+r)!$ w... |
2,381,496 | <p>I am trying to write the explicit formula of all solutions of a linear system in the form :</p>
<p>$Ax=b$ where $A$ is an $m \times n$ matrix ($n$ different from $m$), $x$ is $n$-dimensional vector and $b$ is $m$-dimensional vector.</p>
| mathfan27543 | 279,205 | <p>Hint: isolate for c in both equations: </p>
<p>$c=\frac{ab}4 - a - b$ and $c = \sqrt{a^2+b^2} $. Now isolate for a or b and then substitute back into your equations.</p>
|
949,853 | <p>If $\alpha: [a,b] \rightarrow \mathbb{R}$ is an increasing function, we can define the Riemann-Stieltjes integral $$\int_a^b f d \alpha$$ Does the function $\langle f,g\rangle = \int_a^b fg d\alpha$ define an inner product on the $\mathbb{R}$-space of all Riemann-Stieltjes integrable functions on $[a,b]$ (with respe... | DanZimm | 37,872 | <p>I know this has been answered in the comments but you might be interested in reading up about exactly how this is done over <a href="http://en.wikipedia.org/wiki/Kolmogorov_space#The_Kolmogorov_quotient" rel="nofollow">here</a>. Note that in that article they talk about functions being equal <em>almost everywhere</e... |
1,839,458 | <blockquote>
<p>For any $n\ge2, n \in \mathbb N$ prove that
$$\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$$</p>
</blockquote>
<h3>My work so far:</h3>
<p>1) $$\sqrt{n+1}-\sqrt{n}>\frac1{2\sqrt{n+0.5}}$$</p>
<p>2) $$\sqrt{n+1}-\sqrt... | Nate River | 279,404 | <p>Another proof is this:</p>
<p>Note that
$$
2 = \sqrt{\frac{4n}{n}} = \frac{1}{\sqrt{n}}\sum_{j=0}^{4n-1}\sqrt{j+1}-\sqrt{j}
$$
where the RHS can be expressed as
$$
\frac{1}{\sqrt{n}}\left(\sum_{j=1}^{2n}(\sqrt{2j}-\sqrt{2j-1})+\sum_{j=1}^{2n-1}(\sqrt{2j-1}-\sqrt{2j-2})\right)
$$
Using that the function $f:(0,+\i... |
3,188,966 | <p>Find the number of real solutions of <span class="math-container">$x^7 + 2x^5 + 3x^3 + 4x = 2018$</span>?</p>
<p>What is the general approach to solving this kind of questions? I am interested in the thought process. </p>
<hr>
<p>Few of my thoughts after seeing this question:
since <span class="math-container">$x... | Oscar Lanzi | 248,217 | <p>Render</p>
<p><span class="math-container">$x^7+0x^6+2x^5+0x^4+3x^3+0x^2+4x-2018=0$</span></p>
<p><a href="https://www.mathplanet.com/education/algebra-2/polynomial-functions/descartes-rule-of-sign" rel="nofollow noreferrer">Descartes' Rule of Signs</a> forces exactly one positive root and no negative roots. That... |
3,161,584 | <p>Consider the following matrix</p>
<p><span class="math-container">$A=\begin{bmatrix}0&0&0&0\\a&0&0&0\\0&b&0&0\\0&0&c&0\end{bmatrix}$</span></p>
<p>Which <span class="math-container">$a,b,c$</span> are real numbers</p>
<p>What conditions are required for <span class=... | jmerry | 619,637 | <p>All right, let's clear up some things about the configuration.</p>
<p>First, any planar graph with all faces quadrilaterals (or even polygons) must be bipartite. How do those parts split? If there are six or more vertices in one of the parts, then, since each of those vertices has degree at least <span class="math-... |
402,310 | <p>Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. represented like a column matrix). It was really never explained well in classes.</p>
<p>Here are the things I can't wrap m... | Christopher A. Wong | 22,059 | <p>Actually, I think your intuition is very good here. An "arrow" has two bits of information: its length and direction. When you have a vector $v = [a, b]$, it's describing an arrow that moves $a$ in the horizontal direction and $b$ in the vertical direction. Since this defines a right triangle, this can be converted ... |
2,577,634 | <p>My goal is to compute $$I=\int_{0}^{+∞}\frac{\cos{ax}}{1+x^2}dx$$ where $a>0$. </p>
<p>$$I=\frac{1}{2}\int_{-∞}^{+∞}\frac{\cos{ax}}{1+x^2}dx=\frac{1}{2}Re\bigg(\int_{-∞}^{+∞}\frac{e^{iax}}{1+x^2}dx\bigg)$$. </p>
<p>Let $f(z)=\frac{e^{iaz}}{1+z^2}$.</p>
<p>By Residue Theorem, $\int_{-R}^{R}\frac{e^{iax}}{1+x^2}... | Rebellos | 335,894 | <p>Recall Jordan's Lemma and your solution is done :</p>
<blockquote>
<p>If the only singularities of $F(z)$ are poles, then :
$$\lim_{R\to\infty}\int_{γR} e^{inz}F(z)dz$$
provided that $n>0$ and $|F(z)|\to 0$ as $R\to \infty$.</p>
</blockquote>
<p>Now, you have :</p>
<p>$$\int_{\gamma_R}\frac{e^{iaz}}{1+... |
3,779,736 | <p>Let <span class="math-container">$(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$</span>. Find <span class="math-container">$E\left[\sum \frac{X_i }{X_{(n)}}\right]$</span> where <span class="math-container">$X_{(n)} = \max_i X_i$</span>.</p>
<p>It was suggested to use Basu's Theorem which I am unfamiliar ... | Michael Hardy | 11,667 | <p>Basu's theorem says that a complete sufficient statistic is independent of an ancillary statistic.</p>
<p>To say that <span class="math-container">$X_{(n)} = \max\{X_1,\ldots,X_n\}$</span> is sufficient for this family of distributions (parametrized by <span class="math-container">$b$</span>) means that the conditio... |
23,264 | <p>Let's say that I have a variable $j$ defined by the following formula:
$$j=\frac{n(n+2) + m}{2}$$
where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions:</p>
<ol>
<li>$n\in \left[0,1,2,...\right]$</li>
<li>$m\in\left[-n,n\right]$</li>
<li>$\left(n-m\right)$ even</li>
</ol>
<p>We... | joriki | 6,622 | <p>Since $2j$ depends quadratically on $n$, you basically need the square root of $2j$ to get $n$. To get the limits exactly right, I'd try $n=\lfloor \sqrt{2j+a}+b\rfloor$ and then determine $a$ and $b$ so that everything comes out right. Once you have $n$, it's easy to get $m$.</p>
<p>Update: You can systematically ... |
23,264 | <p>Let's say that I have a variable $j$ defined by the following formula:
$$j=\frac{n(n+2) + m}{2}$$
where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions:</p>
<ol>
<li>$n\in \left[0,1,2,...\right]$</li>
<li>$m\in\left[-n,n\right]$</li>
<li>$\left(n-m\right)$ even</li>
</ol>
<p>We... | Arturo Magidin | 742 | <p>You can take $2j = n(n+2)+m = (n+1)^2 + (m-1)$.</p>
<p>Consider $k^2$ and $(k+1)^2$. The difference between them is $2k+1$; if you allow yourself to add at most $k$ to $k^2$, you "get" all the way to $k^2+k$. If you allow yourself to subtract at most $k$ from $(k+1)^2$, you can "get down" all the way to $k^2+k+1$. ... |
4,218,944 | <p>A very simple question. Saw this formula many places earlier, but how do we prove it?
<span class="math-container">$$ax^2+bx+c=a(x-r_1)(x-r_2)$$</span> Where <span class="math-container">$r_1$</span> and <span class="math-container">$r_2$</span> are the roots of the quadratic.</p>
<p>P.S.- I have seen a "proof&... | julio vasquez | 956,404 | <p>I don't know if that's what you want, but try factoring <span class="math-container">$ax^2+bx+c$</span> and you will have expressions for the roots.</p>
|
1,589,577 | <p>Try to find the solution of the functional equation $$f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$$ with $f'(1)=1/2$.</p>
| Yiorgos S. Smyrlis | 57,021 | <p>I am solving it with the assumption that $f'(0)=1/2$ and $f$ is continuously differentiable in an open neighbourhood containing $0$.</p>
<p>First, we observe that $f(0)=0$. </p>
<p>Then,
$$
\frac{f(x+h)-f(x)}{h}=\frac{\frac{f(x)+f(h)}{1-4f(x)f(h)}-f(x)}{h}
=\frac{f(x)+f(h)-f(x)+4f(h)f^2(x)}{h\big(1-4f(h)f(x)\big)}... |
3,623,856 | <p>Use the definition of a Cauchy sequence to prove that the sequence defined by <span class="math-container">$x_n = \left (\frac{3}{2}\right )^n$</span> is a Cauchy sequence in <span class="math-container">$\mathbb{R}$</span>. </p>
| Mnifldz | 210,719 | <p>This sequence is not Cauchy. Since <span class="math-container">$\mathbb{R}$</span> is complete, then every Cauchy sequence in <span class="math-container">$\mathbb{R}$</span> converges, but you can clearly see that <span class="math-container">$\left (\frac{3}{2}\right )^n \to \infty$</span>.</p>
|
4,172,955 | <p>Find all <span class="math-container">$|z|=1$</span> such that <span class="math-container">$|z^4+4| = \sqrt{5}.$</span></p>
<hr />
<p>I've tried doing
<span class="math-container">$$|z^4+4|^2 = 5 \implies (z^4+4)(\overline{z^4}+4) = 5 \implies |z|^8 + 4(z^4+\overline{z^4})
+11=0,$$</span>
but i'm not sure how to ... | Cameron Williams | 22,551 | <p>Hint: try defining a new variable <span class="math-container">$w = z^4$</span> so that the equation in terms of <span class="math-container">$w$</span> reads (correcting the <span class="math-container">$z^8$</span> term to <span class="math-container">$|z|^8$</span>)</p>
<p><span class="math-container">$$ |w|^2 + ... |
2,903,110 | <p>How do I prove that the function $$f:(0,1)\rightarrow \mathbb R$$ defined by:</p>
<p>$$f(x) = \frac{-2x+1}{(2x-1)^2-1}$$</p>
<p>is onto?</p>
| DeepSea | 101,504 | <p>Alternatively, set $t = 2x-1$ and for each $r \in \mathbb{R}$ we show we can find a number $t$ and then an $x \in (0,1)$ such that$f(x) = r$. We have: $\dfrac{t}{1-t^2}=r$. So this gives $t = r - rt^2\implies rt^2+t -r = 0$. If $r = 0 \implies t = 0\implies x = \dfrac{1}{2}$. If $r \neq 0$, $t = \dfrac{-1\pm \sqrt{1... |
18,421 | <p>I am teaching 4th-grade kids. The topic is Fraction. Basic understanding of a fraction as a part of the whole and as part of the collection is clear to the kids. Several concrete ways exist to teach this basic concept. But when it comes to fraction addition/subtraction I could not find a way that teaches it concrete... | Roar Stovner | 8,357 | <p>Another way to represent addition and subtraction of two fractions is by subdividing rectangles both horizontally and vertically.</p>
<p>Say you want to show how to add <span class="math-container">$\frac{1}{3}$</span> and <span class="math-container">$\frac{2}{5}$</span>. Draw three equal sized rectangles. Subdivid... |
31,451 | <p>I have created a notebook, with functions. The only thing that I want to change in the notebook later is the input file. I therefor want to have a button that runs every input step-by-step so that I do not have to go into every input and press enter for it to update. How could this be done. </p>
| m0nhawk | 2,342 | <p>There is a <a href="http://reference.wolfram.com/mathematica/ref/menuitem/EvaluateNotebook.html" rel="noreferrer"><code>Evaluation -> Evaluate Notebook</code></a> menu for that.</p>
<p>Or, you can set some cells to have <a href="http://reference.wolfram.com/mathematica/tutorial/OptionsForCells.html" rel="norefer... |
1,074,809 | <p>What is a non-decreasing sequence of sets and how come it can have a limit?</p>
<p>It appear in a probability theory book</p>
| Workaholic | 201,168 | <p>By the following</p>
<ul>
<li>$x\mapsto 2x+1$ is always defined.</li>
<li>$x\mapsto \sin x$ is defined for all $x$.</li>
<li>$x\mapsto \sqrt{x}$ is defined if and only if $x\geqslant0$.</li>
</ul>
<p>Your function is a combination of them, thus you should see when the expression inside the radical is positive, i.e... |
3,369,129 | <p>I have the following formulas: </p>
<p><span class="math-container">$$S = 4 \times \tan\left(\frac{180^\circ-a}{2}\right) - \frac{π}{90^\circ} \times (180^\circ-a)$$</span></p>
<p>where:</p>
<p><span class="math-container">$$S \gt 0$$</span></p>
<p><span class="math-container">$$0^\circ \lt a \lt 180^\circ$$</... | Claude Leibovici | 82,404 | <p>Starting from LutzL's answer, we need to find "good" approximation of <span class="math-container">$x$</span> such that
<span class="math-container">$$k=\tan(x)-x \qquad \text{where} \qquad k=\frac S 4$$</span>
For the range <span class="math-container">$ 0 \leq x \leq \frac \pi 4$</span>, a quite good approximation... |
342,491 | <p>How to prove the following:</p>
<p>$a_n = \left\{\left(1+\frac{1}{n}\right)^n\right\}$ is bounded sequence, $ n\in\mathbb{N}$</p>
| Stahl | 62,500 | <p>As Weson Jiang notes, $a_n = \left(n + \frac{1}{n}\right)^n > n^n$, and $n^n\to\infty$ as $n\to\infty$, so $a_n\to\infty$ as well.</p>
<p>You might have wanted to show that $\left\{b_n\right\}_{n\in\Bbb{N}} = \left\{\left(1 + \frac{1}{n}\right)^n\right\}_{n\in\Bbb{N}}$ is a bounded sequence. The binomial theorem... |
2,321,850 | <p>The base is the semicircle $$y=\sqrt{16−x^2},$$
where -4 $\le$ $x$ $\le$ 4. The cross-sections perpendicular to the $x$-axis are squares.
$$\\$$
So far this is what I have:</p>
<p>$\int (Area)\,dx$</p>
<p>$\implies$ $\int \frac{π}{2} ($$r^2$$)\ dx$.</p>
<p>$r$ = $\frac{\sqrt{16−x^2}}{2}$</p>
<p>$\implies$ $ \fra... | Saketh Malyala | 250,220 | <p>Each slice of the solid is a square.</p>
<p>The side length of the square is equal to the height of the function at that point.</p>
<p>So each square has area $(\sqrt{16-x^2})^2=16-x^2$.</p>
<p>You just need to integrate $\displaystyle \int_{-4}^4 (16-x^2)\,dx=\boxed{\frac{256}{3}}$. </p>
<p>Note, the base is th... |
2,559,110 | <p>Prove if $g:\mathbb{R}^2\rightarrow \mathbb{R}$ is a differentiable function, then:</p>
<p>$\frac{d}{dx}g(x,x)=\frac{\partial}{\partial x}g(x,x)+\frac{\partial}{\partial y}g(x,x)$</p>
<p>My work:</p>
<p>$\frac{\partial}{\partial x}g(x,x)=g'(x,x)g(1,1)$</p>
<p><br>$\frac{\partial}{\partial y}g(x,x)=0$</p>
<p><br... | Dr. Sonnhard Graubner | 175,066 | <p>write $$\frac{\sqrt{4x^2+x}-x)(\sqrt{4x^2+x}+x)}{\sqrt{4x^2+x}+x}$$
and then write
$$\frac{x^2(3+\frac{1}{x})}{|x|\left(\sqrt{4+\frac{1}{x}}+1\right)}$$
we get in both cases $\infty$</p>
|
2,559,110 | <p>Prove if $g:\mathbb{R}^2\rightarrow \mathbb{R}$ is a differentiable function, then:</p>
<p>$\frac{d}{dx}g(x,x)=\frac{\partial}{\partial x}g(x,x)+\frac{\partial}{\partial y}g(x,x)$</p>
<p>My work:</p>
<p>$\frac{\partial}{\partial x}g(x,x)=g'(x,x)g(1,1)$</p>
<p><br>$\frac{\partial}{\partial y}g(x,x)=0$</p>
<p><br... | Mark Viola | 218,419 | <blockquote>
<p><strong>I thought it might be instructive to present a very simple way forward that relies on the definition of the limit and an elementary inequality. To that end we proceed.</strong></p>
</blockquote>
<p>Note that for any number $B>0$, however large, we have for $x>0$</p>
<p>$$\begin{align}... |
743,292 | <p>Three fair six-sided dice are thrown and the dice show three different numbers. Find the probability that at least one six is obtained. </p>
<p>Im unsure ofwhat type of question this is, I have tried combinations such as 6C1 over 18C3. </p>
<p>But im not sure if this is correct. Any guidance is much appreciate... | Cm7F7Bb | 23,249 | <p>The number of possible outcomes of this experiment is given by
$$
\frac{6\cdot 5\cdot 4}{3!}.
$$
Outcomes when no $6$ is obtained is given by
$$
\frac{5\cdot 4\cdot 3}{3!}.
$$
The probability that at least one $6$ is obtained
$$
1-\frac{\frac{5\cdot 4\cdot 3}{3!}}{\frac{6\cdot 5\cdot 4}{3!}}=\frac12.
$$</p>
|
3,274,461 | <p>Suppose I have a recurrence relation like : </p>
<p><span class="math-container">\begin{equation}
f(n)=\begin{cases}
2*f(n-1) , & \text{n%2 = 1}.\\
f(n-1) + 1, & \text{n%2 = 0}.
\end{cases}
\end{equation}</span></p>
<p><span class="math-container">\begin{equation}
f(0) = f(1) = 1
\end{equation... | John Omielan | 602,049 | <p>I'm not sure if this type of problem has a special name. It's just basically a function definition split between <span class="math-container">$2$</span> different types of values, in this case depending on whether <span class="math-container">$n$</span> is odd or even.</p>
<p>The way I would approach trying to solv... |
3,382,119 | <p><span class="math-container">$x\cdot\begin{bmatrix}0\\-3\\1\end{bmatrix}$</span> +
<span class="math-container">$y\cdot\begin{bmatrix}-5\\2\\-4\end{bmatrix}$</span> +
<span class="math-container">$z\cdot\begin{bmatrix}-20\\-1\\-13\end{bmatrix}$</span>
=<span class="math-container">$\begin{bmatrix}0\\0\\0\end{bmatrix... | Bernard | 202,857 | <p>Supposin your RREF is exact (which it is <em>not</em>), there remains to interpret it as an equivalent linear system of equations:
<span class="math-container">$$\begin{cases}
x+3z=0\\y+4z=0
\end{cases}\implies \begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}-3z\\-4z\\\quad z\end{bmatrix}=z\begin{bmatrix}-3\\-4\\... |
3,382,119 | <p><span class="math-container">$x\cdot\begin{bmatrix}0\\-3\\1\end{bmatrix}$</span> +
<span class="math-container">$y\cdot\begin{bmatrix}-5\\2\\-4\end{bmatrix}$</span> +
<span class="math-container">$z\cdot\begin{bmatrix}-20\\-1\\-13\end{bmatrix}$</span>
=<span class="math-container">$\begin{bmatrix}0\\0\\0\end{bmatrix... | J. W. Tanner | 615,567 | <p><span class="math-container">$x\cdot\begin{bmatrix}0\\-3\\1\end{bmatrix}$</span> +
<span class="math-container">$y\cdot\begin{bmatrix}-5\\2\\-4\end{bmatrix}$</span> +
<span class="math-container">$z\cdot\begin{bmatrix}-20\\-1\\-13\end{bmatrix}$</span>
=<span class="math-container">$\begin{bmatrix}0\\0\\0\end{bmatrix... |
13,274 | <p>I have searched stackoverflow (and comparable pages) for quite a while now (got redirected from there to this specialized stack), and I surrender. I am trying to evaluate an expression that is small in the end numerically. <br></p>
<p>Example:</p>
<pre>Log[Log[Log[6^5^4^3^2^1]]]=12.9525...<br></pre>
<p>WolframAlp... | Rojo | 109 | <p>An alternative could be</p>
<pre><code>Block[{Power, Log},
Log[Log[Log[6^5^4^3^2^1]]] // PowerExpand]
</code></pre>
<blockquote>
<p>Log[262144 Log[5] + Log[Log[2] + Log[3]]]</p>
</blockquote>
<pre><code>% // N
</code></pre>
<blockquote>
<p>12.9525</p>
</blockquote>
<p>Still gives overflows, it's equivalent... |
1,096,453 | <p>Let $X$ be an inner product space. If $x_n\to x$, $y_n\to y$ (in norm), then $(x_n,y_n)\to(x,y)$ (in modulus).</p>
| Pp.. | 203,995 | <p>Add and subtract $(x,y_n)$ and use Cauchy's and triangle inequality many times.</p>
<p>$$\begin{align}|(x,y)-(x_n,y_n)|&=|(x,y)-(x,y_n)+(x,y_n)-(x_n,y_n)|\\&\leq |(x,y-y_n)|+|(x-x_n,y_n)|\\&\leq ||x||\cdot||y-y_n||+||x-x_n||\cdot||y_n||\\&\leq||x||\cdot||y-y_n||+||x-x_n||\cdot(||y||+\epsilon)\end{al... |
1,096,453 | <p>Let $X$ be an inner product space. If $x_n\to x$, $y_n\to y$ (in norm), then $(x_n,y_n)\to(x,y)$ (in modulus).</p>
| Surb | 154,545 | <p>I'm not sure what you mean by "modulus" but other definition of the norm on $X \times Y$ could probably be done with very similar proofs.</p>
<p>Let $\|\cdot\|_X,\|\cdot\|_Y$ be norms on $X$ and $Y$ and $\|\cdot\|_{X\times Y}:X \times Y \to \Bbb R: (u,v)\mapsto \sqrt{\|u\|_X^2+\|v\|^2_2}$. Let us suppose that $(x_n... |
4,527,455 | <p>My goal is to find all values of "a" so that the circle <span class="math-container">$x^2 - ax + y^2 + 2y = a$</span> has the radius 2</p>
<p>The correct answer is: <span class="math-container">$a = -6$</span> and <span class="math-container">$a = 2$</span></p>
<p>I tried solving it by doing this:<br/>
<sp... | binbni | 1,011,566 | <p><span class="math-container">$\frac{a^2+4a+4}{4}$</span> is not a radius.
Actually, it is the square of radius.</p>
<p>So, you should solve <span class="math-container">$\frac{a^2+4a+4}{4}=2^2$</span></p>
<p>And its solution is a=-6, a=2</p>
|
56,804 | <p>I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. </p>
<p>I know that if you have a 3-cycle and a 2-cycle operating on three elements, ... | zyx | 14,120 | <p>The question implicitly assumes the base field is $Q$. The Galois group of a polynomial only exists with reference to a specified field containing the coefficients. It is not enough to assume that $X^3 - 2$ is irreducible, because the Galois group is cyclic of order 3 (that is, a proper subgroup of S_3 rather than... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.