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 |
|---|---|---|---|---|
199,199 | <p>Suppose a box contains 5 white balls and 5 black balls.</p>
<p>If you want to extract a ball and then another:</p>
<p>What is the probability of getting a black ball and then a black one?</p>
<p>I think that this is the answer:</p>
<p>Let $A:$ get a black ball in the first extraction, $B:$ get a black ball in th... | André Nicolas | 6,312 | <p>It is good to check an answer for plausibility. <strong>Given</strong> that the first ball is black, surely the probability that the second ball is black cannot be $\dfrac{8}{9}$!</p>
<p>In this case, it is the <strong>conditional probability</strong> that is obvious. Note that $\Pr(B|A)=\dfrac{4}{9}$. And what we ... |
1,435,665 | <p>Matrix $\mathbf{D}$ is a full rank diagonal matrix. $\mathbf{ADA}^T=\mathbf{D}$.</p>
<p>Can we conclude that $\mathbf{A}^2=\mathbf{I}_n$? (edited from $\mathbf{A}=\pm\mathbf{I}$)</p>
<p>It's almost sure that some one has asked this question before, but I cannot find proper keywords to search for that post.</p>
| DirkGently | 88,378 | <p>By taking the derivative of $y^2 + xy -3x = 5$ with respect to $t$, we get
$$2y\frac{dy}{dt}+x\frac{dy}{dt}+\frac{dx}{dt}y-3\frac{dx}{dt}=0.$$
By substitution,
$$-2-1-\frac{dx}{dt}-3\frac{dx}{dt}=0,$$
implying that
$$\frac{dx}{dt}=-\frac34\cdot$$</p>
|
3,506,340 | <p>I have no idea how to calculate <span class="math-container">$z_x+z_y$</span> at a point <span class="math-container">$\left( \frac{\pi +3}{3}, \frac{\pi+1}{2}\right)$</span>, if <span class="math-container">$z=uv^2$</span> and <span class="math-container">$x=u+sinv$</span>, <span class="math-container">$y=v+cosu$</... | Quanto | 686,284 | <p>Differentiate <span class="math-container">$u+\sin v=x$</span> and <span class="math-container">$v+\cos u=y$</span> with respect to <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, respectively,</p>
<p><span class="math-container">$$u_x+\cos v\> v_x=1,\>\>\>\>\>... |
316,016 | <p>Could you recommend any approachable books/papers/texts about matroids (maybe a chapter from somewhere)? The ideal reference would contain multiple examples, present some intuitions and keep formalism to a necessary minimum.</p>
<p>I would appreciate any hints or appropriate sources.</p>
| davidlowryduda | 9,754 | <p>Dover has released a cheap version of Welsh's <a href="http://rads.stackoverflow.com/amzn/click/0486474399">Matroid Theory</a> - can't argue with the price. On the other hand, I've heard better things about Oxley's <a href="http://rads.stackoverflow.com/amzn/click/0199603391">Matroid Theory</a>.</p>
<p>(Apparently ... |
5,363 | <p>There is something in the definition of the <a href="http://en.wikipedia.org/wiki/Free_product" rel="nofollow noreferrer">free product</a> of two groups that annoys me, and it's this "word" thing:</p>
<blockquote>
<p>If <span class="math-container">$G$</span> and <span class="math-container">$H$</span> are groups... | Dylan Wilson | 423 | <p>Well, I think that the usual definition of the free product is just a slightly less formal version of what you've written, which, by the way, is certainly correct. I haven't seen it completely written out like that before but maybe others have.</p>
<p>Meanwhile, one can avoid the whole messy construction and prove ... |
5,363 | <p>There is something in the definition of the <a href="http://en.wikipedia.org/wiki/Free_product" rel="nofollow noreferrer">free product</a> of two groups that annoys me, and it's this "word" thing:</p>
<blockquote>
<p>If <span class="math-container">$G$</span> and <span class="math-container">$H$</span> are groups... | Tsuyoshi Ito | 926 | <p>This is not an answer to your questions, but I guess I would point out that the “words” in the definition in Wikipedia can be given a structure: consider the free monoid freely generated by the set <span class="math-container">$G \sqcup H$</span>, and just call its elements <em>words in G and H</em>. Now you can in... |
3,317,088 | <blockquote>
<p>Let
<span class="math-container">\begin{align*}
\mathcal{P} : &\min_{x}~& (x + a)^2 \\
& s.t. & x + b \leq 0
\end{align*}</span></p>
</blockquote>
<p>where <span class="math-container">$ a, b \in \mathbb{R} $</span>. I would like to find <span class="math-contain... | k.stm | 42,242 | <p>No, it doesn’t. The point is that any chain of primes in <span class="math-container">$R / \mathfrak p$</span> lifts to a chain of primes in <span class="math-container">$R$</span> <em>above</em> <span class="math-container">$\mathfrak p$</span>. But since <span class="math-container">$R$</span> is an integral domai... |
30,402 | <p>The envelope of parabolic trajectories from a common launch point is itself a parabola.
In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally celebrated with fireworks.</p>
<p>If the launch point is the origin, and the trajectory starts off at angle $\theta$ and... | Joseph O'Rourke | 6,094 | <p>I found another article to supplement those to which Andrey linked:
Eugene I Butikov, "<a href="http://iopscience.iop.org/0143-0807/24/4/101" rel="nofollow">Comment on 'The envelope of projectile trajectories'</a>,"
<em>Eur. J. Phys.</em> <strong>24</strong> L5-L9, 2003.
He also explains the expanding-circles viewpo... |
2,906,350 | <p>I have already consulted V.K. Rohatgi, and it has an example where it takes $Y=X^a$ where $a>0$ but the domain of $X$ is positive real values.<br>
Even the theorem for transformation of continuous random values restricts the derivative of $Y$ w.r.t. $X$ to be positive or negative for the entire domain of $X$ wher... | Hongyu Wang | 381,246 | <p>We have</p>
<p>\begin{align*}
P(Y \leq y) &= P(X^3 \leq y)\\
&= P(X \leq y^\frac{1}{3})
\end{align*}</p>
<p>Now we take the derivative in this case with $F$ being the CDF of X and f being the pdf of X.</p>
<p>$$F(y^\frac{1}{3})' = \frac{1}{3}f(y^\frac{1}{3})y^{-\frac{2}{3}}$$</p>
<p>Thus you just subsitu... |
719,056 | <p>I'm writing a Maple procedure and I have a line that is "if ... then ..." and I would like it to be if k is an integer (or if k^2 is a square) - how would onw say that in Maple? Thanks!</p>
| Abhishek Verma | 94,307 | <p>There is a $C$ which <strong>looks like</strong> $A/B$ (suppose) in some german notation. And now in some rule you denote some value by $C$ and others by $A$ and $B$ and $C = A/B$ is the rule. You won't say you broke the notation $C$ into $A/B$ because it was looking so.</p>
<p>You will deal more such cases in inte... |
689,546 | <p>I know this problem involves using Cantor's theorem, but I'm not sure how to show that there are more subsets of an infinite enumerable set than there are positive integers. It seems like a lot of these problems are really the same problem, but they require some unique and creative thought to get them just right. An... | P Vanchinathan | 28,915 | <p>Given an enumerable set $S=\{x_1,x_2,\ldots\}$, and an infinite sequence of 0's and 1's, $\{b_1,b_2,\ldots\}$, we can associate a subset of $S$ consisting of those $x_i$ for which $b_i=1$. This is a bijection between colection of subsets of $S$ and infinite binary sequences. These binary sequences can be through of ... |
2,078,737 | <p>I will gladly appreciate explanation on how to do so on this matrix:</p>
<p>$$
\begin{pmatrix}
i & 0 \\
0 & i \\
\end{pmatrix}
$$</p>
<p>I got as far as calculating the eigenvalues and came up with $λ = i$.
when trying to find the eigenvectors I came up with the $0$ matrix.<... | Tsemo Aristide | 280,301 | <p>$x\rightarrow 1^{-}$, $x\pi/2\rightarrow \pi/2^{-}$, </p>
<p>$sin(x\pi/2)\rightarrow 1$, $cos(x\pi/2)\rightarrow 0^+$ so the limit is $+\infty$.</p>
|
1,810,582 | <p>Statement 1: </p>
<blockquote>
<p>If $f :A\mapsto B$ & if $X,Y$ are subsets of $B$, then the inverse image of union of two sets $X$ & $Y$ is the union of inverse images of $X$ & $Y$.</p>
</blockquote>
<p>Statement 2:</p>
<blockquote>
<p>Let $f:A\mapsto B$ be a one-to-one correspondence from $... | Henno Brandsma | 4,280 | <p>I'd fill in the details of "it's easy to check" steps. They are crucial here (you must use the triangle inequality somewhere in the proof, as the theorem does not hold for non-metric spaces). </p>
<p>Also state what the supposed countable base is, beforehand. Then show it is indeed countable, and then do this proof... |
1,749,284 | <p>As part of my homework i've the following question:</p>
<p>The tangent line $ L $ is crossing the graph of $ y = ax^3 + bx $ at point $ x = x_0 $, find another point where the tangent-line $L$ is crossing the graph. Define $ a = 1$ and $b = 0$.</p>
<p>Second part of the question is to graph $y = x^3$ and show the ... | SimpleProgrammer | 1,102,366 | <p>Tupper's self-referential formula (when suitably modified) can be absolutely used to plot any two coloured image of arbitrary height and width. Thus, indeed, you can take a screenshot on a 5K iMac and make it monochromatic and you will be able to graph it provided you have modified the formula suitably. The modifica... |
1,932,599 | <p>Find the solution of the differential equation that satisfies the given initial condition <span class="math-container">$$~y'=\frac{x~y~\sin x}{y+1}, ~~~~~~y(0)=1~$$</span></p>
<p>When I integrate this function I get
<span class="math-container">$$y+\ln(y)= -x\cos x + \sin x + C.$$</span></p>
<p>Have I integrated ... | nmasanta | 623,924 | <p>Given that <span class="math-container">$$~y'=\frac{x~y~\sin x}{y+1}$$</span>
<span class="math-container">$$\implies \frac{dy}{dx}=\frac{x~y~\sin x}{y+1}$$</span>
<span class="math-container">$$\implies \frac{y+1}{y}~dy=x\sin x~dx$$</span>
<span class="math-container">$$\implies \left(1+\frac{1}{y}\right)~dy=x\sin ... |
2,354,383 | <p>Why doesn't a previous event affect the probability of (say) a coin showing tails?</p>
<p>Let's say I have a <strong>fair</strong> and <strong>unbiased</strong> coin with two sides, <em>heads</em> and <em>tails</em>.</p>
<p>For the first time I toss it up the probabilities of both events are equal to $\frac{1}{2}$... | farruhota | 425,072 | <p>Imagine one tosses $1,000,000$ fair coins at once. Hence the outcomes of the coins are independent and then the numbers of heads and tails will be approximately (if not exactly) equal.</p>
|
3,758,536 | <p>Theorem 3.29: If <span class="math-container">$p>1$</span>,<br />
<span class="math-container">$$
\sum_{n=2}^{\infty}\frac{1}{n(\log\ n)^p}
$$</span>
converges; if <span class="math-container">$p\leq1$</span>, the series diverges.</p>
<p>Proof: The monotonicity of the logarithmic function implies that <span class... | Kavi Rama Murthy | 142,385 | <p>Product of two non-negative increasing functions is increasing. Any positive power of an increasing function is increasing. So <span class="math-container">$n \log n$</span> and <span class="math-container">$n \log^{p}n$</span> are increasing. If <span class="math-container">$f$</span> is increasing and <span class=... |
2,081,612 | <p>I'd like to evaluate the following integral:</p>
<p>$$\int \frac{\cos^2 x}{1+\tan x}dx$$</p>
<p>I tried integration by substitution, but I was not able to proceed. </p>
| DXT | 372,201 | <p>$\displaystyle \int \frac{\cos^2 x}{1+\tan x}dx = \int\frac{\cos^3 x}{\sin x+\cos x}dx = \frac{1}{2}\int\frac{(\cos^3 x+\sin ^3 x)+(\cos^3 x-\sin ^3 x)}{\cos x+\sin x}dx$</p>
<p>$\displaystyle = \frac{1}{4}\int (2-\sin 2x)dx+\frac{1}{4}\int\frac{(2+\sin 2x)(\cos x-\sin x)}{\cos x+\sin x}dx$</p>
<p>put $\cos x+\sin... |
592,963 | <p>Find $x^4+y^4$ if $x+y=2$ and $x^2+y^2=8$</p>
<p>So i started the problem by nothing that $x^2+y^2=(x+y)^2 - 2xy$ but that doesn't help!</p>
<p>I also seen that $x+y=2^1$ and $x^2+y^2=2^3$ so maybe $x^3+y^3=2^5$ and $x^4+y^4=2^7$ but i think this is just coincidence</p>
<p>So how can i solve this problem?</p>
<p... | Carl Love | 104,071 | <p>Solve the given equations simultaneously to obtain $x=1-\sqrt 3, y=1+\sqrt 3$ or vice versa. Then just compute $x^4+y^4$ directly.</p>
|
1,647,517 | <p>I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it?
\begin{equation}
\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx=\sqrt{2\pi}e^{-\frac{n^{2}\pi^{2}}{2}},
\end{equation}
If anybody already knows it is welcome,... | gniourf_gniourf | 51,488 | <p>Here's a funny one:</p>
<p>Define the function $f$ as
$$f:\mathbb{R}\longrightarrow\mathbb{R},\ a\longmapsto\int_{-\infty}^{+\infty}\mathrm{e}^{-x^2/2}\cos(a x)\,\mathrm{d}x.$$
It is well-known that the improper integral defining $f$ is convergent. We can apply the differentiation theorem to differentiate this impr... |
1,647,517 | <p>I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it?
\begin{equation}
\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx=\sqrt{2\pi}e^{-\frac{n^{2}\pi^{2}}{2}},
\end{equation}
If anybody already knows it is welcome,... | Jack D'Aurizio | 44,121 | <p>Another chance is given by the expansion of $\cos(\pi n x)$ as a Taylor series:
$$ \cos(\pi n x) = \sum_{m\geq 0}\frac{(-1)^m (\pi n)^{2m} x^{2m}}{(2m)!}\tag{1} $$
together with the fact that:
$$ \int_{\mathbb{R}} x^{2m} e^{-x^2/2}\,dx = \int_{0}^{+\infty} x^{m-1/2} e^{-x/2}\,dx = 2^{m+1/2}\cdot\Gamma\left(m-\frac{1... |
27,666 | <p>I have reading a book myself; more than 90 % of the book is ok to me; but I have some problems on it. Is it ok to ask my questions in a related chat-room? Because I believe that they are so easy, and it seems shameful to me to ask such an absurd question?</p>
| samerivertwice | 334,732 | <p>I frequently have this same problem but I have learnt to ask the question anyway. Provided the question or equivalent doesn't already exist, often the answer is given in a comment rather than an answer, as users are embarrassed to answer such trivial questions. But inevitably somebody later needs an answer to the sa... |
2,913,156 | <p>I'm working through the definitions given in <strong>2.18</strong> in Baby Rudin, and I was wondering how to prove that <span class="math-container">$E = [0,1] \cap \Bbb{Q}$</span> is closed in <span class="math-container">$\Bbb{Q}$</span>.</p>
<p>Clearly, the set <span class="math-container">$[0,1] \cap \Bbb{R}$</... | fleablood | 280,126 | <p>I'm sorry to say I couldn't follow your argument that it is intuitive $[0,1]$ is closed in $\mathbb R$ at all.</p>
<p>$[0,1]$ is closed (in $\mathbb R$) because all its limit points are in in $[0,1]$. That is; If $p \in \mathbb R$ and for every $\epsilon > 0$ it will always be that the interval $(p-\epsilon, p+... |
897,955 | <p>For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof</p>
<p>I'm following a book and it just uses this, it doesn't say anything about the function, so I've not assumed it's one-to-one. I've spent to much time on this (as in it should be 4 lines for both ways at most) so I hope you ... | Troy Woo | 74,973 | <p>$$
\Rightarrow: U\subset f^{-1}(f(U))\subset f^{-1}(V)
$$
$$
\Leftarrow: f(U)\subset f(f^{-1}(V))\subset V
$$
Just read through (maybe more than once) the first chapter of James Munkres, <strong>Topology</strong>. Its the best to me. </p>
|
1,270,107 | <p>How many real roots does the below equation have?</p>
<p>\begin{equation*}
\frac{x^{2000}}{2001}+2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0
\end{equation*}</p>
<p>A) 0 B) 11 C) 12 D) 1 E) None of these</p>
<p>I could not come up with anything.</p>
<p>(Turkish Math Olympiads 2001)</p>
| wythagoras | 236,048 | <p>We have that $x^{2000} \geq 0$, because squares are nonnegative.</p>
<p>Further, we have $(x-\frac{1}{2}\sqrt{\frac{5}{3}})^2 \geq 0$. This gives $x^2-\sqrt{\frac{5}{3}}x+\frac{5}{12} \geq 0$</p>
<p>Therefore $2 \sqrt{3}x^2-2\sqrt{5}x+\frac{10}{12}\sqrt{3} \geq 0$</p>
<p>Therefore $2 \sqrt{3}x^2-2\sqrt{5}x+\sqrt{... |
489,528 | <p>I need to check for convergence here. Why is every example completely different and nothing which I've learned from the examples before helps me in the next? Hate it!</p>
<p>$$ \sum_{n\geq1} \frac{n^{n-1}}{n!} $$</p>
<p>so I tried the ratio test and I got</p>
<p>$$ \frac{1}{1+\frac{1}{n}} $$
which converges to $... | John | 23,668 | <blockquote>
<p>Why is every example completely different and nothing which I've
learned from the examples before helps me in the next?</p>
</blockquote>
<p>Because in mathematics you need to think in principles, not in rote repetition. Every problem is new, and you have to determine which of all the various fact... |
4,074,031 | <p>You have been teaching Dennis and Inez math using the Moore method for their entire lives, and you're currently deep into topology class.</p>
<p>You've convinced them that topological manifolds aren't quite the right object of study because you "can't do calculus on them," and you've managed to motivate th... | David E Speyer | 448 | <p><span class="math-container">$\def\RR{\mathbb{R}}$</span>I have in fact tried to tackle this problem when I teach higher differential forms. Here are some of the things I try.</p>
<p><b>Before we do differential forms or manifolds</b> For functions <span class="math-container">$f : \RR^n \to \RR^p$</span>, introduce... |
207,382 | <p>I was interested in exploiting the Terminal` package, as demonstrated in <a href="https://mathematica.stackexchange.com/questions/13961/plotting-graphics-as-ascii-plots">this post</a>, to show some ASCII plots straight through the terminal of my Raspberry Pi Model 3 B+. However, it doesn't seem that I can get a plot... | Lukas Lang | 36,508 | <p>Here's an approach to fix the code behind <code>Terminal`</code> - Execute the following code to restore the functionality of the package:</p>
<pre><code><< Terminal`
ExportString["", "TTY"];
DownValues@System`Convert`BitmapDump`ExportTTY = DownValues@System`Convert`BitmapDump`ExportTTY /.
{
TextStyle... |
3,034,374 | <p>Two cyclists start from the same place to ride in the same direction.A starts at noon with a speed of 8km/hr and B starts at 2pm with a speed of 10km/hr.At what times A and B will be 5km apart ?
My thought process:
As A starts early at 12 so it will have already covered 16km(8*2).
so S relative=V relative*t or say 1... | Shubham Johri | 551,962 | <p>By the time <span class="math-container">$B$</span> starts, <span class="math-container">$A$</span> is ahead by <span class="math-container">$16$</span> km. The velocity of <span class="math-container">$B$</span> with respect to <span class="math-container">$A, v_{BA}=v_B-v_A=2$</span> kmph. Therefore, the distance ... |
2,260,975 | <p>Problem: Let $\mu : \mathcal{B}(\mathbb{R}^k) \rightarrow \mathbb{R}$ be a nonnegative and finitely additive set function with $\mu(\mathbb{R}^k) < \infty $.
Suppose that
\begin{equation}
\mu(A) = \sup \{ \mu(K) : K \subset A, K \text{ compact}\}
\end{equation}
for each $A \in \mathcal{B}(\mathbb{R}^k)$.
Then $... | Nathanael Skrepek | 423,961 | <p>$\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\irun}[1]{_{#1 \in \N}}$</p>
<p>Unfortunately math.stackexchange hast problems with the <code>\intertext</code> command.</p>
<p>This is a very detailed proof for your problem</p>
<p>$\Omega = \R^n$,
$\mathfrak{A} = \mathfrak{B}(\R^n)$</p>
<p>At... |
244 | <p>I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a Hilbert scheme to be smooth?</p>
| David E Speyer | 297 | <p>The Hilbert scheme of n points on a 3-fold is not smooth for n sufficiently large, but the exact value of sufficiently large is unknown. See the chapter on Hilbert schemes in "Combinatorial Commutative Algebra", by Miller and Sturmfels.</p>
|
244 | <p>I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a Hilbert scheme to be smooth?</p>
| mdeland | 397 | <p>Here is yet another example of a smooth Hilbert scheme. Let $X$ be a smooth degree 3 hypersurface in projective space of dimension $n \geq 3$ (say, over an algebraically closed field), and let $H$ be the Hilbert scheme of lines on $X$ (i.e., corresponding to Hilbert polynomial $t + 1$). </p>
<p>The tangent space ... |
1,990,105 | <p>Presume that $(x_n)$ is a sequence s. t.</p>
<p>$|x_n-x_{n+1}| \le 2^{-n}$ for all $n \in \mathbb N$</p>
<p>Prove that $x_n$ converges.</p>
<p>What I've tried to think: since $2^{-n}$ converges to 0, and the difference between the terms $x_n$ and $x_{n+1}$ is smaller or equal to it, then $x_n$ must be a cauchy s... | Yiorgos S. Smyrlis | 57,021 | <p>Let $\varepsilon>0$.</p>
<p>There exists an $N$, such that $2^{-N}<\varepsilon$. (Say $N=\lfloor-\log_2\varepsilon\rfloor+1$.)</p>
<p>Now let $m,n>N$, then
$$
\lvert x_m-x_n\rvert\le \sum_{k=n}^{m-1}\lvert x_{k+1}-x_k\rvert\le
\sum_{k=n}^{m-1}2^{-k}<2^{-(n-1)}\le 2^{-N}<\varepsilon.
$$</p>
|
125,834 | <p>Is there a general formula for the number of distinguishable arrangements of length $\ell$ of a sequence of size $s$, given the repeated elements? So,</p>
<p>For example, would there be a general formula to solve the problem:</p>
<p>How many distinguishable 5-letter arrangements are there of the word "PROBABILITY"... | Jair Taylor | 28,545 | <p>This may indeed be "overkill", but there is a fairly simple general formula for the exponential generating function. Let <span class="math-container">$n_i$</span> be the number of times the <span class="math-container">$i$</span>th letter is repeated - the order we put the letters in doesn't matter of co... |
2,672,497 | <p>$$\lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}}$$ I unsuccessfully tried to find two different Riemann Sums converging to the same value close to the given sum so I could use the Squeeze Theorem. Is there any other way to solve this?</p>
| Mark Viola | 218,419 | <p>It is easy to see that</p>
<p>$$ \begin{align}
\left |\sum_{k=1}^n \frac{1}{n+k+k/n^2}-\sum_{k=1}^n \frac1{n+k}\right|&=\frac1{n^2}\sum_{k=1}^n \frac{k}{(n+k+k/n^2)(n+k)}\\\\
&\le \frac1{n^2}\sum_{k=1}^n \frac1k\tag 1
\end{align}$$ </p>
<p>Then, using $\sum_{k=1}^n\frac1k =\gamma+\log(n) +O\left(\frac1n\ri... |
496,882 | <p>If I have an experiment that has $1000$ trials, and $10$% of the time there is an error, what is the approximate probability that I will have $125$ failures? I figured out that $\mu =100$ and $\sigma =9.4868$. I'm trying to approximate with a normal distribution, since with this many trials, and the fact that each e... | robjohn | 13,854 | <p>You are computing the value for going over $83$ standard deviations above the mean. The chance of going above that are approximately
$$
\frac1{\sqrt{2\pi}}\frac1{83}e^{-83^2/2}
$$
which is very small; $\sim2.5\times10^{-1494}$. The chance of getting less than 83 standard deviations is therefore approximately $1$.</p... |
1,018,672 | <blockquote>
<p><span class="math-container">$$\int_0^{\infty} \frac{1}{x^3-1}dx$$</span></p>
</blockquote>
<p>What I did:</p>
<p><span class="math-container">$$\lim_{\epsilon\to0}\int_0^{1-\epsilon} \frac{1}{x^3-1}dx+\lim_{\epsilon\to0}\int_{1+\epsilon}^{\infty} \frac{1}{x^3-1}dx$$</span></p>
<hr />
<p><span class="ma... | Timbuc | 118,527 | <p>Using partial fractions directly I think it is simpler:</p>
<p>$$\int\frac1{x^3-1}dx=\frac13\left(\int\frac1{x-1}-\frac{x+2}{x^2+x+1}\right)dx=$$</p>
<p>$$\frac13\left(\log(x-1)-\frac12\int\frac{2x+1}{x^2+x+1}dx-\frac32\int\frac1{\left(x+\frac12\right)^2+\frac34}dx\right)=$$</p>
<p>$$=\frac13\log(x-1)-\frac12\log... |
607,917 | <p>Any Help solving this question ?</p>
<blockquote>
<p>a) Find ONE solution <span class="math-container">$\overline x\in\Bbb Z/325\Bbb Z$</span> such that <span class="math-container">$x^2\equiv-1\pmod{325}$</span>. (Hint: CRT and lifting.)</p>
<p>b) How many solutions <span class="math-container">$\overline x$</span... | Tim Ratigan | 79,602 | <p>$325=5^2\cdot 13$ so let's solve $x^2\equiv -1\pmod{25}$, $x^2\equiv -1\pmod{13}$.</p>
<p><strong>Case $1$: $\mod{25}$</strong></p>
<p>If $x^2\equiv -1\pmod{25}$, then $x^2\equiv -1\pmod 5$ so $x\equiv\pm2\pmod 5$</p>
<p>$$(5k+2)^2=25k^2+20k+4\equiv 20k+4\equiv-1\pmod{25}\implies 4k\equiv -1\pmod 5\\\implies k\eq... |
1,304,971 | <p>Using a triangular facet approximation of a sphere based on <a href="http://paulbourke.net/geometry/circlesphere/" rel="nofollow">Sphere Generation by Paul Bourke</a>.</p>
<p>We take an octahedron and bisect the edges of its facets
to form 4 triangles from each triangle.</p>
<p><code>
/\ ... | user7530 | 7,530 | <p>The key is the Euler characteristic formula
$$V-E+F = 2.$$</p>
<p>You've already calculated that $F=8\cdot 4^N$. You also know that all of the facets are triangles; that means that each triangle has three half-edges or $3F=2E$. That means
$$2V - F = 4$$
or
$$V = 2 + 4^{N+1}.$$</p>
|
2,564,256 | <p>The definition of general complex log for any non-zero complex number $z$ is
$$Log(z)=\log|z|+i[\arg(z)+2m\pi], m\in \mathbb{Z}$$</p>
<p>With this, if $n\in \mathbb{N}$ then
$Log(z^{1/n})=\frac{1}{n} Log(z)$ holds for all non-zero complex number $z$. </p>
<p>I verified this for $Log(i^{1/2})=\frac12 Log(i)$ succ... | Gaurang Tandon | 89,548 | <p>Let $z^3-i=0$, then, by cube roots of unity, its roots are $-i, -i\omega, -i\omega^2$, for $\omega=e^{i2\pi/3}$ the cube root of unity. These values translate to $e^{i3\pi/2}, e^{i\pi/6}, e^{i5\pi/6}$.</p>
<p><strong>Hint:</strong> I am showing this proof for $z=e^{i3\pi/2}$, but the procedure is same for the other... |
2,424,722 | <blockquote>
<p>Let a,b,c be real numbers such that $a+2b+c=4$. What is the value of $\max(ab+bc+ac)$</p>
</blockquote>
<p>My attempt:</p>
<p>Squaring both the sides: </p>
<p>$a^2 +4b^2+c^2+2ac+4bc+4ab=16$</p>
<p>Then I tried factoring after bringing 16 to LHS but couldn't. It's not even a quadratic in one variab... | Donald Splutterwit | 404,247 | <p>Lagrange multipliers will give the solution
\begin{eqnarray*}
L=ab+bc+ca+ \lambda (a+2b+c-4)
\end{eqnarray*}
Differentiating gives
\begin{eqnarray*}
b+c+ \lambda =0 \\
c+a +2\lambda =0 \\
a+b+ \lambda =0 \\
\end{eqnarray*}
Add these equation together and substitute to get $ \lambda = \frac{b-4}{2}$ now substitute th... |
2,424,722 | <blockquote>
<p>Let a,b,c be real numbers such that $a+2b+c=4$. What is the value of $\max(ab+bc+ac)$</p>
</blockquote>
<p>My attempt:</p>
<p>Squaring both the sides: </p>
<p>$a^2 +4b^2+c^2+2ac+4bc+4ab=16$</p>
<p>Then I tried factoring after bringing 16 to LHS but couldn't. It's not even a quadratic in one variab... | dxiv | 291,201 | <p>From the given condition $\,2b=4-a-c\,$, then:</p>
<p>$$\require{cancel}
\begin{align}
2(ab+bc+ca) &= 2b(a+c)+2ac \\
&= (a+c)(4-a-c)+2ac= \\
&= 4a - a^2 - \bcancel{ac} + 4c - \bcancel{ac} - c^2 + \bcancel{2ac} = \\
&= \color{red}{8} -a^2 + 4a \color{red}{-4} - c^2 +4c \color{red}{-4} = \\
&=... |
3,744,610 | <p>The following puzzle was introduced by the psychologist Peter Wason in 1966, and is one of the
most famous subject tests in the psychology of reasoning.</p>
<blockquote>
<p>Four cards are placed on the table in front of you. You are told
(truthfully) that each has a letter printed on one side and a digit on
the othe... | Deepak | 151,732 | <p>Your answer (not the book's) is correct. You are given a proposition that is a unidirectional implication <span class="math-container">$vowel \implies odd$</span>. You need to test this by turning over any vowel card(s) you see to confirm the reverse side(s) is(are) odd(s).</p>
<p>But every implication also implies ... |
3,694,661 | <p>I was stuck on a problem from <a href="https://rads.stackoverflow.com/amzn/click/com/0821804308" rel="nofollow noreferrer" rel="nofollow noreferrer" title="Quite a Fun Recreational Math Book">Mathematical Circles: Russian Experience</a>, which reads as follows:</p>
<blockquote>
<p><em>Prove that the number <span ... | Anas A. Ibrahim | 650,028 | <p>I guess that the power is chosen because when working with modular arithmetic, one would directly consider <strong>euler's theorem</strong> (which is a generalization of fermat's little theorem) and see if it helps:
<span class="math-container">$$a^{\phi(n)} \equiv 1 \pmod{n}$$</span>
If <span class="math-container"... |
1,621,792 | <p>I'm stuck on this problem:</p>
<blockquote>
<p>Let <span class="math-container">$C\subset \mathbb{R}^n$</span> be closed and unbounded.</p>
<p>Suppose <span class="math-container">$f:C\to\mathbb{R}^m$</span> is continuous and such that <span class="math-container">$\lim_{x\to\infty} f(x)$</span> exists and is finit... | ncmathsadist | 4,154 | <p>Let $\epsilon > 0$ and $L$ be the limit of $f$ at $\infty$. Choose $M$ so that $|x| > M \Rightarrow |f(x) - L| <\epsilon$. Notice that $\{x\in C| |x|\le M\}$ is compact so $f$ is uniformly continous there. Can you do the rest?</p>
|
1,621,792 | <p>I'm stuck on this problem:</p>
<blockquote>
<p>Let <span class="math-container">$C\subset \mathbb{R}^n$</span> be closed and unbounded.</p>
<p>Suppose <span class="math-container">$f:C\to\mathbb{R}^m$</span> is continuous and such that <span class="math-container">$\lim_{x\to\infty} f(x)$</span> exists and is finit... | Wolfy | 217,910 | <p>Suppose $f$ is continuous but not uniformly continuous on $[a,b]$. Then, there exists an $\varepsilon_0 > 0$ such that $$|x - y| < \delta \ \Rightarrow \ |f(x) - f(y)| < \varepsilon_0 \ \ \text{is false} \ \forall \delta$$ For $\delta = \frac{1}{N}$, we can find two points $x_n,y_n\in [a,b]$ such that $$|x... |
131,579 | <p>I need some help solving this problem.</p>
<p>A man is about to perform a random walk. He is standing a distance of 100 units from a wall. In his pocket, he has 10 playing cards: 5 red and 5 black.</p>
<p>He shuffles the cards and draws the top card.</p>
<p>If he draws a red card, he moves 50 units (half the di... | Brian M. Scott | 12,042 | <p>Some hints: </p>
<ol>
<li><p>$792$ is even, so $13xy45z$ is even; this tells you something about the possibilities for $z$. In fact, it’s divisible by $8$, which means that the three-digit number $45z$ has to be divisible by $8$.</p></li>
<li><p>$792$ is divisible by $9$, so $13xy45z$ is as well; do you know the ea... |
1,770,656 | <p>In lecture I was given the following reasoning:</p>
<p>There are 7 options for the first character, 6 options for the last character, and 9! combinations for the 9 characters in between. Then you divide out the repeats -- 4 I's, 4 S's, and 2 P's. So the answer is 7 * 6 * 9! / (4! * 4! * 2!) = 13230</p>
<p>My own r... | Ben | 313,305 | <p>You only subtracted the combinations both beginning and ending with I. There are cases that the letter either start with I or end with I.</p>
<p>So you want to calculate the number of combinations that start with I: $\frac{10!}{4!*3!*2!}$</p>
<p>and the number of combinations that end with I: $\frac{10!}{4!*3!*2!}... |
889,728 | <p>Evaluate $$\int_0^{1/2}x^3\arctan(x)\,dx$$</p>
<p>My work so far:
$x^3\arctan(x) = \sum_{n=0}^\infty(-1)^n \dfrac{x^{2n+4}}{2n+1}$</p>
<p>$$\int_0^{1/2}x^3\arctan(x)\,dx = \sum_{n=0}^\infty \frac{(-1)^{n+1}(1/2)^{2n+4}}{2n+1}$$</p>
<p>The desired accuracy needed is to four decimal places, that is were I am stuck.... | David H | 55,051 | <p><strong>Hint:</strong> You might consider first integrating parts:</p>
<p>$$\begin{align}
\int_{0}^{1/2}x^3\arctan{(x)}\,\mathrm{d}x
&=\frac14x^4\arctan{(x)}\bigg{|}_{0}^{1/2}-\frac14\int_{0}^{1/2}\frac{x^4}{1+x^2}\,\mathrm{d}x\\
&=\frac{1}{64}\arctan{\left(\frac12\right)}-\frac14\int_{0}^{1/2}\frac{x^4}{1+... |
96,957 | <p>Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was wondering if there are natural examples of stacks that arise in arithmetic geometry or number theory.</p>
<p>To me, ... | Zack Wolske | 18,086 | <p>One big recent example would be Lafforgue's proof of the Langlands correspondence for $GL_n$ of function fields (<a href="http://arxiv.org/abs/math.NT/0212399" rel="noreferrer">http://arxiv.org/abs/math.NT/0212399</a>), which uses stacks of schtukas. It is similar to Drinfel'd's proof for $GL_2$, but with the moduli... |
390,644 | <p>I'm trying to solve this recurrence relation:</p>
<p>$$
a_n = \begin{cases}
0 & \mbox{for } n = 0 \\
5 & \mbox{for } n = 1 \\
6a_{n-1} - 5a_{n-2} + 1 & \mbox{for } n > 1
\end{cases}
$$</p>
<p>I calculated generator function as:
$$
A = \frac{31x - 24x^2}{1 - 6x + 5x^2} + \frac{x^3}{(... | vonbrand | 43,946 | <p>Use Wilf's technique from "generatingfunctionology". Define $A(z) = \sum_{n \ge 0} a_n z^n$, and write:
$$
a_{n + 2} = 6 a_{n + 1} - 5 a_n + 1 \qquad a_0 = 0, a_1 = 5
$$
Multiply by $z^n$ and add for $n \ge 0$, which gives:
$$
\frac{A(z) - a_0 - a_1 z}{z^2} = 6 \frac{A(z) - a_0}{z} - 5 A(z) + \frac{1}{1 - z}
$$
Solv... |
1,771,279 | <p>Let $B=\{x \in \ell_2, ||x|| \leqslant 1\}$ - Hilbert ball $X \subset B$ - open convex connected set in Hilbert ball, $\bar{X}$ - closure of $X$. $F: \bar{X} \to \bar{X}$ - continuous map that holomorphic into each point of $X$. Is it true in general, that $F$ has fixed point?</p>
| Yiorgos S. Smyrlis | 57,021 | <p>Define $f: \overline{X}\to \overline{X}$, where $X=\{x\in B: 1/2<\|x\|<1\}$, as
$$
f(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots).
$$
No fixed point for $f$ is $B$.</p>
|
1,541,800 | <p>I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$</p>
<p>And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial, then:
$$ P(A)=\begin{bmatrix}
P(a) & P'(a) \... | Martin R | 42,969 | <p>If $$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$
then by induction you can prove that
$$ A^n = \begin{bmatrix}
a^n & n a^{n-1} \\
0 & a^n
\end{bmatrix} \tag 1
$$
for $n \ge 1 $. If $f$ can be developed into a power series
$$
f(z) = \sum_{n=0}^\in... |
4,101,068 | <p>Can two random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> have the covariance matrix</p>
<p><span class="math-container">$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}? $</span></p>
<p>The matrix is positive definite if <span class="math-container">$xy<0... | Cm7F7Bb | 23,249 | <p>The eigenvalues of the matrix
<span class="math-container">$$
\begin{pmatrix}
0&-1\\
-1&0
\end{pmatrix}
$$</span>
are <span class="math-container">$-1$</span> and <span class="math-container">$1$</span>. The eigenvalues of a positive semidefinite matrix are non-negative. Hence, this matrix is not positive se... |
1,956,855 | <p>I'm doing some math work involving proofs, and one of the definitions is:</p>
<p>|a| = -a when a < 0</p>
<p>Isn't the absolute value of a, positive a no matter what a is in the beginning? Am I looking at this wrong? Could use an explanation.</p>
| arkeet | 50,739 | <p>$\renewcommand\sgn{\operatorname{sgn}}$The general idea is this: For any nonzero real number $a$, we can decompose it as $s \cdot t$, where $s = +1$ or $-1$ is a sign, and $t > 0$ is a positive number. I'll call these $s = \sgn(a)$ and $t = |a|$. (Here $\sgn$ is called the <a href="https://en.wikipedia.org/wiki/S... |
4,580,478 | <p>So I'm currently reading about signed measures in "Real analysis" by Folland. In it, he defines a signed measure as follows.</p>
<p><em>Let <span class="math-container">$(X,\mathcal{M})$</span> be a measurable space. Then a signed measure <span class="math-container">$\nu$</span> is a function from <span c... | Simon SMN | 712,612 | <p>Okay, I've done some digging and I think I've formulated my original hesitancy and resolved it.</p>
<p>Let <span class="math-container">$(X,\mathcal{M})$</span> be a measurable space and suppose <span class="math-container">$\nu':\mathcal{M}\to[-\infty,\infty]$</span>. Suppose further that</p>
<p>(i) <span class="ma... |
16,797 | <p>Is there any good way to approximate following integral?<br>
$$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$<br>
$\mu$ is between $0$ and $0.25$, the problem is in $\sigma$ which is always positive, but it can be arbitrarily small.<br>
I was trying to expa... | whuber | 1,489 | <p>A standard way to get a good approximation for integrals that "look" Gaussian is to evaluate the Taylor series of the logarithms of their integrands through second order, expanding around the point of maximum value thus (continuing with @Ross Millikan's substitution):</p>
<p>$$\eqalign{
&\log\left(\sqrt{y}\cd... |
1,531,171 | <p>if $fg$ and $f$ are differentiable at $a$, must $g$ be differentiable at $a$? If not, what condition is needed to imply that $g$ be differentiable at $a$?</p>
<p>I realized that people asked similar question before, like
"If $f+g$ and $f$ are differentiable at $a$, must $g$ be differentiable at $a?$" It can be easi... | Thomas Andrews | 7,933 | <p>The most commonly referenced non-differentiable function is $g(x)=|x|$. Let $f(x)=x$. Then $f$ is differentiable, as is $fg$.</p>
<p>Obviously, if $f$ and $fg$ are differentiable at $a$ and $f(a)\neq 0$ then $g=(fg)/f$ is differentiable at $a$ by the usual quotient rule.</p>
|
4,206,994 | <p>Let <span class="math-container">$F(X, Y)$</span> be an irreducible polynomial of <span class="math-container">$\mathbb{C}[X, Y]$</span> with <span class="math-container">$F(0, 0) = 0$</span> and non-singular there, and let <span class="math-container">$(X(t), Y(t))$</span> be a local analytical parametrization of t... | orangeskid | 168,051 | <p>The result you need is the following: in a irreducible complex variety every open subset in the standard topology is Zariski dense. Therefore, if two irreducible complex varieties <span class="math-container">$X$</span>, <span class="math-container">$X'$</span> (say inside some bigger space) are such that <span cl... |
1,985,849 | <p>I am really stumped by this equation.
$$e^x=x$$
I need to prove that this equation has no real roots. But I have no idea how to start.</p>
<p>If you look at the graphs of $y=e^x$ and $y=x$, you can see that those graphs do not meet anywhere. But I am trying to find an algebraic and rigorous proof. Any help is appre... | Community | -1 | <p>Let $f(x) = e^x - x$. $f'(x) = e^x -1 \geq 0$ and hence $f$ is increasing. Thus for any $x > 0$, $f(x) > f(0)$ and hence $e^x > x+1$ and $f(x) = 0$ has no solution when $x > 0$. For $x <0$, $e^x >0$ and $x < 0$ and hence $e^x \neq x$.</p>
|
1,985,849 | <p>I am really stumped by this equation.
$$e^x=x$$
I need to prove that this equation has no real roots. But I have no idea how to start.</p>
<p>If you look at the graphs of $y=e^x$ and $y=x$, you can see that those graphs do not meet anywhere. But I am trying to find an algebraic and rigorous proof. Any help is appre... | Fred | 380,717 | <ol>
<li><p>For $x \le 0$ we have $e^x >0$, thus no solution of the above equation exists in $(-\infty, 0)$</p></li>
<li><p>If $x>0$, then $e^x=1+x+\frac{x^2}{2!}+\ldots>x$. Therefore: no solution of the above equation exists in $(0, \infty)$.</p></li>
</ol>
|
1,595,243 | <p>Let $f:\mathbb R \to \mathbb R$ be a function such that $|f(x)|\le x^2$, for all $x\in \mathbb R$. Then, at $x=0$, is $f$ both continuous and differentiable ?</p>
<p>No idea how to begin. Can someone help?</p>
| Jimmy R. | 128,037 | <p>You need to use the <a href="https://en.wikipedia.org/wiki/Squeeze_theorem">squeeze theorem</a> on the given condtition $|f(x)|\le x^2$ in order to prove that $f$ is both continuous and differentiable at $x=0$:</p>
<ol>
<li>Continuity: You know that $|f(x)|\le x^2$. Substituting $x=0$ you find that $|f(0)|\le 0$ or... |
1,764,729 | <blockquote>
<p>Let $x(t) : [0,T] \rightarrow \mathbb{R}^n$ be a solution of a differential equation
$$
\frac{d}{dt} x(t) = f(x(t),t).
$$
In addition we have functions $E :\mathbb{R}^n \rightarrow \mathbb{R}$ and $h:\mathbb{R}^{n+1}\rightarrow \mathbb{R}$ such that
$$
\frac{d}{dt}{E(x(t))} = h(x(t),t) E(x(t))... | Vincent | 332,815 | <p>This is a partial answer (but may put someone on the right track for a full one)</p>
<p>In the case $E$ and $h$ are $C^{\infty}$ (infinitely derivable, like it is often the case in physics) and if there is a constant $M$ that bounds ALL of the derivatives of $E$:</p>
<p>Let us assume $E(t_0) = 0$ for some $t_0$</p... |
3,474,926 | <p>I came across this question:</p>
<p>Find an isomorphism from the group of orientation preserving isometries of the plane to some subgroup of <span class="math-container">$GL_{2}(\mathbb C)$</span>.</p>
<p>I'm having trouble with finding such isomorphism. Mainly, I'm having trouble with finding some representation ... | Moishe Kohan | 84,907 | <p>Hint: Think of affine complex linear-fractional transformations of the form <span class="math-container">$z\mapsto az+b$</span>, <span class="math-container">$|a|=1$</span>, and the epimorphism <span class="math-container">$GL(2, {\mathbb C})\to PGL(2, {\mathbb C})$</span>, the group of complex linear-fractional tra... |
505,617 | <p>Let $a,b,c$ be nonnegative real numbers such that $a+b+c=3$, Prove that</p>
<blockquote>
<p>$$ \sqrt{\frac{a}{a+3b+5bc}}+\sqrt{\frac{b}{b+3c+5ca}}+\sqrt{\frac{c}{c+3a+5ab}}\geq 1.$$</p>
</blockquote>
<p>This problem is from <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=555716">http:... | user1793963 | 96,691 | <p>Since $a+b+c=3$ you can reduce $f(a,b,c)$ to $f(3-b-c, b, c)$. With only 2 variables you can make a 3D plot and see if it gets below $1$ for $b+c\le3$. Not an elegant proof, but it works. If nothing else it will at least show you where the edge cases are.</p>
|
4,373,216 | <p>Find power series solution for <span class="math-container">$y''-xy=0$</span> close to <span class="math-container">$x=0$</span>.</p>
<p><span class="math-container">$\sum_{n=2}^\infty n(n-1)a_nx^{n-2}-x\sum_{n=0}^\infty a_nx^n=0$</span></p>
<p>Then,
<span class="math-container">$a_{n+2}(n+2)(n+1)-a_{n-1}=0 \implies... | Michael Burr | 86,421 | <p>Hint: Break the integral into pieces depending on its domains of definition. In your linked example,</p>
<p><span class="math-container">$$
\int_{-1}^3 f(x)dx=\int_{-1}^0f(x)dx+\int_0^1f(x)dx+\int_1^2f(x)dx+\int_2^3f(x)dx.
$$</span>
Substituting formulas, you would have
<span class="math-container">$$
\int_{-1}^00d... |
782,427 | <p><strong>I - First doubt</strong> : free variables on open formulas . </p>
<p>I'm having a hard time discovering what different kinds of variables in an open formula of fol are refering to., For example, lets take some open formulas : </p>
<p>1: $"x+2=5"$<br>
2: $"x=x"$<br>
3: $"x+y = y+x"$<br>
4: $"x+2 = ... | jdc | 7,112 | <p>I don't know if this will help, but here's the formalism I use.</p>
<p>Let a first-order language $L$ be given, an open formula $\varphi(x,y)$ in the language of $L$, with only free variables $x$ and $y$, and an $L$-structure $M$. Let $X$ be the collection of all variables you are using.</p>
<p>There is no way to ... |
1,918,674 | <p>For what value $k$ is the following function continuous at $x=2$?
$$f(x) = \begin{cases}
\frac{\sqrt{2x+5}-\sqrt{x+7}}{x-2} & x \neq 2 \\
k & x = 2
\end{cases}$$</p>
<p>I was thinking about multiplying the numerator by it's conjugate, but that makes the denominator very messy, so I don't rlly know what to d... | ajotatxe | 132,456 | <p>Do as you were thinking. Don't expand the denominator and cancel out the factor $x-2$.</p>
|
519,093 | <p>Let $C[0,1]$ be the set of all continuous real-valued functions on $[0,1]$.</p>
<p>Let these be 3 metrics on $C$.</p>
<p>$p(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|$</p>
<p>$d(f,g)=(\int_0^1|f(t)-g(t)|^2dt)^{1/2}$</p>
<p>$t(f,g)=\int_0^1|f(t)-g(t)|dt$</p>
<p>Prove that for every $f,g\in C$, the following holds $t(f,g)\... | kel c. | 58,740 | <p>you can invoke if $x \le y$ and $z \ge 0$, then $zx \le zy$ and the transitivity of the ordered fields. </p>
<p>try $xy$ and compare that to $xx$ and $yy$.</p>
|
1,045,941 | <p>Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it?
I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} \dfrac{\log(x+h)-\log(x)}{h}$$
but this led to no developments.</p>
| Aditya Hase | 190,645 | <p>$$\begin{align}
f'(x)&=\lim_{h\rightarrow 0} \dfrac{f(x+h)-f(x)}{h}\tag{1}\\
&=\lim_{h\rightarrow 0} \dfrac{\log(x+h)-\log(x)}{h}\tag{2}\\
&=\lim_{h\rightarrow 0} \dfrac{\log\left(\frac{x+h}{x}\right)}{h}\tag{3}\\
&=\lim_{h\rightarrow 0} \dfrac{\log\left(1+\frac{h}{x}\right)}{h}\tag{4}\\
&=\lim_{... |
2,583,232 | <p>Prove that $\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}=\dfrac{2}{5}$</p>
<p>$$\forall \epsilon>0 \ \ :\exists N(\epsilon)\in \mathbb{N} \ \ \text{such that} \ \ \forall n >N(\epsilon) \ \ \text{we have} |\dfrac{-2}{25n^2+5}|<\epsilon$$
$$ |\dfrac{-2}{25n^2+5}|<\epsilon \\ |\dfrac{2}{25n^2+5}|<\epsi... | Community | -1 | <p>Given the limit $\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}$ you have to group by $n^2$. It becomes:
$$\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}=\lim_{n \to \infty} \dfrac{n^2\cdot 2}{n^2\cdot\left(5+\frac 1{n^2}\right)}=\lim_{n \to \infty} \dfrac{2}{\left(5+\frac 1{n^2}\right)}=\dfrac 25$$</p>
<p>Because we know that... |
1,051,335 | <p>Purely out of interest, I wanted to try and construct a sequence of differentiable functions converging to a non-differentiable function. I began with the first non-differentiable function that sprung to my mind, namely
\begin{align}
&f:\mathbb{R}\to\mathbb{R}\\
&f(x)=|x|.
\end{align}
After some testing I co... | Alain | 138,095 | <p>Consider the sequence of differentiable functions $h_n(x) = x^{1+\frac{1}{2n-1}}$ defined on $[-1,1]$ and note
$$
\lim_{n\rightarrow\infty} h_n(x) =
x\lim_{n\rightarrow\infty} x^\frac{1}{2n-1}=
|x|,\qquad \forall x\in[-1,1].
$$</p>
|
1,977,817 | <p>Find the perimeter of the square shown, I don't know how to solve it do I use pythagorean theorem?
<a href="https://i.stack.imgur.com/cdzq7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cdzq7.jpg" alt="enter image description here"></a></p>
| DonAntonio | 31,254 | <p>Hints (for you to complete):</p>
<p>1) Both diagonals of a square meet at the circumscribing circle's center.</p>
<p>2) If a square's side's length is $\;x\;$ , then any of its two diagonals' length is $\;\sqrt2\,x\;$</p>
|
1,977,817 | <p>Find the perimeter of the square shown, I don't know how to solve it do I use pythagorean theorem?
<a href="https://i.stack.imgur.com/cdzq7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cdzq7.jpg" alt="enter image description here"></a></p>
| scott | 330,966 | <p>Try going this direction: draw another radius of the circle to an adjacent corner of the square. Now you have a right triangle, the length of the legs is $3\sqrt{2}$. Can you find the length of the hypotenuse? This is one of the sides of the square; multiply this length by 4 to find the perimeter.</p>
|
3,720,012 | <blockquote>
<p>(Feller Vol.1, P.241, Q.35) Let <span class="math-container">$S_n$</span> be the number of successes in <span class="math-container">$n$</span> Bernoulli trials. Prove
<span class="math-container">$$E(|S_n-np|) = 2vq b(v; n, p) $$</span>
where <span class="math-container">$v$</span> is the integer such ... | StubbornAtom | 321,264 | <p>The hint given is not quite correct as mentioned by @PepeSilvia.</p>
<p>For any random variable <span class="math-container">$X$</span> with finite mean <span class="math-container">$\mu$</span>, we have</p>
<p><span class="math-container">\begin{align}
\mathbb E[|X-\mu|]&=\mathbb E[|X-\mu|I_{X>\mu}]+\mathbb ... |
1,329,214 | <p>I'm having difficulty solving a linear algebra problem:<br>
Let $A,B,C,D$ be real $n \times n$ matrices. Show that there is a non-zero $n \times n$ matrix $X$ such that $AXB$ and $CXD$ are both symmetric. </p>
<p>There is an accompanying hint:<br>
Show that the set of all matrices $X$ for which $AXB$ is symmetric... | Matt Samuel | 187,867 | <p>Hint: The product $AXB$ has entries that can be expressed as linear combinations of the entries of $X$, where the coefficients come from the entries of $A$ and $B$. The symmetry condition imposes equations on the entries, so we have a linear system of equations with unknowns the off-diagonal entries of $X$. How many... |
1,410,048 | <p>I am unsure of how to proceed about finding the solution to this problem. $$\sum_{i=6}^8(\sum_{j=i}^8 (j+1)^2)$$</p>
<p>Obviously the last step is not to difficult, but the fact that the lower limit for the summation in brackets is i I am not sure how to solve this. In classes so far we have only really dealt with ... | Hypergeometricx | 168,053 | <p>As pointed out in the first answer, it would be easier to compute it by simple expansion since $i$ can take on only $3$ values. </p>
<p>However since you asked about reindexing this is a possible approach:</p>
<p>$$\sum_{i=6}^8\sum_{j=i}^8(j+1)^2=\sum_{i=7}^9\sum_{j=i}^9 j^2=\sum_{7\le i\le j\le 9} j^2=\sum_{j=7}... |
3,914,365 | <p>I'm pretty stuck with this exercise. Hope somebody can help me:</p>
<p>show that
<span class="math-container">$$
\sum_{n=2^{k}+1}^{2^{k+1}} \frac{1}{n} \geq\left(2^{k+1}-2^{k}\right) \frac{1}{2^{k+1}}=\frac{1}{2}
$$</span>
and use this to show that the harmonic series is divergent.</p>
<p>First I can't figure out ho... | xpaul | 66,420 | <p>In fact
<span class="math-container">$$
\sum_{n=2^{k}+1}^{2^{k+1}} \frac{1}{n}=\frac{1}{2^{k}+1}+\frac{1}{2^{k}+2}+\cdots \frac{1}{2^{k}+2^{k}}\ge \frac{1}{2^{k}+2^k}+\frac{1}{2^{k}+2^k}+\cdots+\frac{1}{2^{k}+2^k}=(2^{k+1}-2^k)\frac{1}{2^{k+1}}=\frac12.
$$</span></p>
|
3,843,807 | <p>Let <span class="math-container">$d_1$</span> be a metric on <span class="math-container">$X$</span>. If there exists a strictly increasing, continuous, and subadditive <span class="math-container">$ f:\mathbb {R} \to \mathbb {R} _{+}$</span> such that <span class="math-container">$d_{2}=f\circ d_{1}$</span>. Then <... | Hamilton | 549,321 | <p>It is extremely simple once you read carefully the content in the <a href="https://www.wikiwand.com/en/Equivalence_of_metrics#/Topological_equivalence" rel="nofollow noreferrer">link</a> you post.</p>
<p>Note that there is an equivalent definition of equivalence of metric:</p>
<blockquote>
<p>the open balls "ne... |
262,773 | <p>In the curve obtained with following</p>
<pre><code> Plot[{1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1}, PlotRange -> {0, 1},
GridLines -> {{0.35, 0.94}, {}}]
</code></pre>
<p>how can one fill the top and bottom with different colors or patterns such that two regions are perfectly visible in a black-n-white printout?<... | MarcoB | 27,951 | <p>Since you mentioned that your final product will be in black and white (or perhaps grayscale), I recommend using hatched fillings instead of colors:</p>
<pre><code>Plot[
Evaluate@ConstantArray[1/(3 x*Sqrt[1 - x^2]),2],
{x, 0.35, .94},
PlotRange -> {{0, 1}, {0, 1}},
GridLines -> {{0.35, 0.94}, {}},
G... |
3,898,327 | <p>I tried expanding <span class="math-container">$n$</span> parentheses such as <span class="math-container">$a b^{-1} a$</span> and then tried to replace the second <span class="math-container">$a$</span> with <span class="math-container">$a^2 a^{-1}$</span> but didn't know how to proceed further.</p>
<p>Also, is it ... | Robert Lewis | 67,071 | <p>I do it <em>via</em> a simple inductive argument:</p>
<p>The trivial identity</p>
<p><span class="math-container">$aba^{-1} = aba^{-1} \tag 1$</span></p>
<p>provides the <span class="math-container">$k = 1$</span> base case; for general <span class="math-container">$k$</span> we assume</p>
<p><span class="math-conta... |
3,898,327 | <p>I tried expanding <span class="math-container">$n$</span> parentheses such as <span class="math-container">$a b^{-1} a$</span> and then tried to replace the second <span class="math-container">$a$</span> with <span class="math-container">$a^2 a^{-1}$</span> but didn't know how to proceed further.</p>
<p>Also, is it ... | Community | -1 | <p>The inner adjacent <span class="math-container">$aa^{-1}$</span> terms cancel out, and this is fairly trivial: <span class="math-container">$(aba^{-1})^n=aba^{-1}aba^{-1}\dots aba^{-1}=ab^na^{-1}$</span>.</p>
|
494,338 | <p>This is gre preparation question of data interpretation,
Distribution of test score among students(Score range -> total % of students)</p>
<blockquote>
<p>0-65 -> 16<br/>
65-69 -> 37<br/>
70-79 -> 25<br/>
80-89 -> 14 <br/>
90-100 -> 8<br/></p>
</blockquote>
<hr>
<p>Question is that</p>
<p>Which of t... | QED | 91,884 | <p>You are completely wrong. The median can be calculated as follows, write down the observations in a increasing order including multiplicities, and then choose the middlemost value, you wrote down the frequency of the corresponding classes and chose the median of the frequencies, not the median of the given data.</p>... |
3,379,447 | <p>I am currently studying for an exam in Mathematical research tools. One concept that has come up again and again throughout the course is continuity and although the definition of Cauchy Continuity or Delta Epsilon continuity was discussed, we were never shown how to actually prove that a function is continuous. The... | Mohammad Riazi-Kermani | 514,496 | <p>If you want to prove that the function <span class="math-container">$f(x)$</span> is continuous at <span class="math-container">$x=a$</span> you need to show that for a given <span class="math-container">$\epsilon >0$</span> you can find a <span class="math-container">$\delta>0$</span> such that if <span cla... |
2,513,509 | <p>If $f,g,h$ are functions defined on $\mathbb{R},$ the function $(f\circ g \circ h)$ is even:</p>
<p>i) If $f$ is even.</p>
<p>ii) If $g$ is even.</p>
<p>iii) If $h$ is even.</p>
<p>iv) Only if all the functions $f,g,h$ are even. </p>
<p>Shall I take different examples of functions and see?</p>
<p>Could anyone ... | José Carlos Santos | 446,262 | <p>Does the limit$$\lim_{h\to0}\frac{\arcsin h-\arctan h}{\sin h-\tan h}$$exist? If it does, it will be equal to your limit. But\begin{align}\lim_{h\to0}\frac{\arcsin h-\arctan h}{\sin h-\tan h}&=\lim_{h\to0}\frac{\left(h+\frac{h^3}6+\frac{3h^5}{40}+\cdots\right)-\left(h-\frac{h^3}3+\frac{h^5}5+\cdots\right)}{\left... |
121,791 | <p>Now I'm trying to use </p>
<pre><code>Show[plota, Epilog->Inset[inset, Scaled->[{0.2, 0.7}]]]
</code></pre>
<p>operation to insert an inset of a figure, where plota is an Overlay of two plots</p>
<pre><code>plota = Overlay[{plotb, plotc}];
</code></pre>
<p>and then I cannot be successfully inserted in beca... | Jason B. | 9,490 | <p>You can just make everything an <a href="http://reference.wolfram.com/language/ref/Inset.html" rel="nofollow noreferrer"><code>Inset</code></a>, this way you don't even need an underlying <code>Graphics</code> object to build on top of. Using the plots from Mr. Wizard's post,</p>
<pre><code>insetA = Overlay[{plotB... |
2,369,274 | <p>More specifically, I am trying to solve the problem in Ravi Vakil's notes:</p>
<blockquote>
<p>Make sense of the following sentence: "The map <span class="math-container">$$\mathbb{A}^{n+1}_k \setminus \{0\} \rightarrow \mathbb{P}^n_k$$</span> given by <span class="math-container">$$(x_0,x_1,..x_n) \rightarrow ... | Ross Millikan | 1,827 | <p>First you can just compute $f$ at $0,1,2$ getting $-2,1,-2$ so clearly A and C are wrong. Because the quadratic term is negative the function will go to $-\infty$ as $x$ gets large in either direction, so it will have some absolute maximum, and D is wrong. You can compute the derivative at $0,2$ and find it is non... |
2,369,274 | <p>More specifically, I am trying to solve the problem in Ravi Vakil's notes:</p>
<blockquote>
<p>Make sense of the following sentence: "The map <span class="math-container">$$\mathbb{A}^{n+1}_k \setminus \{0\} \rightarrow \mathbb{P}^n_k$$</span> given by <span class="math-container">$$(x_0,x_1,..x_n) \rightarrow ... | zhw. | 228,045 | <p>The functions agree at $1$ so $f$ is continuous. Verify that $f'(x) > 0$ to the left of $1,$ and $f'(x) <0$ to the right of $1.$ It follows that $f$ has an absolute max at $1.$</p>
|
2,006,565 | <p>I was solving a question to find $\lambda$ and this is the current situation:</p>
<p>$$\begin{bmatrix}
5 & 10\\
15 & 10
\end{bmatrix}
=\lambda\begin{bmatrix}2\\3\end{bmatrix}$$</p>
<p>My algebra suggested that I should divide the right-hand matrix with the left-hand one to find $\lambda$. Then after a few ... | lab bhattacharjee | 33,337 | <p>Let $\cos^{-1}\sqrt x=y\implies x=(\cos y)^2=\dfrac{1+\cos2y}2\implies dx=-\sin2y\ dy$</p>
<p>and $\cos2y=2x-1$</p>
<p>As $\sqrt x\ge0,0\le y\le\dfrac\pi2,\sin2y=2\sqrt{x(1-x)}$</p>
<p>$$I=\int(a-\cos^{-1}\sqrt x)^2\ dx=-\int(a-y)^2\sin2y\ dy$$</p>
<p>Integrating by parts, </p>
<p>$$I=(a-y)^2\int(-\sin2y)\ dy-\... |
828,148 | <p>In Euclidean geometry, we know that SSS, SAS, AAS (or equivalently ASA) and RHS are the only 4 tools for proving the congruence of two triangles. I am wondering if the following can be added to that list:-</p>
<p>If ⊿ABC~⊿PQR and the area of ⊿ABC = that of ⊿PQR, then the two triangles are congruent.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ {\rm mod}\ 5\!:\,\ \color{#c00}{3^2\equiv\, -1}\ \Rightarrow\ 3^{2k}\equiv (\color{#c00}{3^2})^k\equiv (\color{#c00}{-1})^k\ $ [$\,\equiv -1\ $ if $\,k\,$ is odd, as in your case]</p>
|
4,333,062 | <p>I stumbled upon the following identity
<span class="math-container">$$\sum_{k=0}^n(-1)^kC_k\binom{k+2}{n-k}=0\qquad n\ge2$$</span>
where <span class="math-container">$C_n$</span> is the <span class="math-container">$n$</span>th Catalan number. Any suggestions on how to prove it are welcome!</p>
<p>This came up as a ... | Yiorgos S. Smyrlis | 57,021 | <p>If
<span class="math-container">$$
\sqrt{2k+1}-\sqrt{2k-5}=4
$$</span>
the
<span class="math-container">$$
6=(2k+1)-(2k-5)=(\sqrt{2k+1}-\sqrt{2k-5})(\sqrt{2k+1}+\sqrt{2k-5})
=4(\sqrt{2k+1}+\sqrt{2k-5}).
$$</span>
Hence we obtain the <em>system</em>
<span class="math-container">$$
\sqrt{2k+1}-\sqrt{2k-5}=4 \\
\sqrt{2... |
1,466,281 | <p>Let $X$ and $Y$ be two absolutely continuous random variables on some probability space having joint distribution function $F$ and joint density $f$. Find the joint distribution and the joint density of the random variables $W=X^2$ and $Z=Y^2$.</p>
<p>I tried the following. I know that $$F(w,y)=P[W \leq w,Z \leq z]... | Michael Hardy | 11,667 | <p>You want this:
$$
f_{W,Z}(w,z) = \frac{\partial^2}{\partial w\,\partial z} \int_{-\sqrt{z}}^{\sqrt{z}}\int_{-\sqrt{w}}^{\sqrt{w}} f(x,y) \, dx \, dy.
$$</p>
<p>Use the chain rule:</p>
<p>Write $u= \sqrt z$ so that $u^2 = z$ and $2u\,du = dz$ and then you have
\begin{align}
\frac{df}{dz} & = \frac{df}{du}\cdot\... |
2,787,250 | <p>I don't understand why there are constant $A,B,C$ s.t.
$$\frac{1}{(x-1)(x-2)^2}=\frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-2)^2}.$$</p>
<p>I now how to compute $A,B,C$, but I don't understand how someone though to do this. In what this is natural ?</p>
| Surb | 154,545 | <p>It's just Bézout's Identity. There are $A,B\in\mathbb R$ s.t. $$A(x-1)+B(x-2)=1.$$
Then $$\frac{1}{(x-1)(x-2)^2}=\frac{A(x-1)+B(x-2)}{(x-1)(x-2)^2}=\frac{A}{(x-2)^2}+\frac{B}{(x-1)(x-2)}=\frac{A}{(x-2)^2}+\frac{B(A(x-1)+B(x-2))}{(x-1)(x-2)}=\frac{A}{(x-2)^2}+\frac{BA}{(x-2)}+\frac{B^2}{(x-1)}.$$</p>
<hr>
<h2>Edit<... |
2,787,250 | <p>I don't understand why there are constant $A,B,C$ s.t.
$$\frac{1}{(x-1)(x-2)^2}=\frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-2)^2}.$$</p>
<p>I now how to compute $A,B,C$, but I don't understand how someone though to do this. In what this is natural ?</p>
| G Cab | 317,234 | <p>Expanding upon request my comment above, I am going to briefly expose the statement that Partial Fractions
naturally proceed from the chain<br>
Residue Theorem $\to$ <a href="https://en.wikipedia.org/wiki/Laurent_series" rel="nofollow noreferrer">Laurent Series</a> $\to$ <a href="https://en.wikipedia.org/wiki/Mittag... |
2,403,851 | <p>Here is Proposition 3 (page 12) from Section 2.1 in <em>A Modern Approach to Probability</em> by Bert Fristedt and Lawrence Gray. </p>
<blockquote>
<p>Let $X$ be a function from a measurable space $(\Omega,\mathcal{F})$ to another measurable space $(\Psi,\mathcal{G})$. Suppose $\mathcal{E}$ is a family of subsets... | lhf | 589 | <p>All log functions differ by a multiplicative constant. Therefore, notations like $O(n \log n)$ are unambiguous, because the $O$ absorbs multiplicative constants.</p>
|
3,533,023 | <p>We have <span class="math-container">$x_1+x_2+...+x_k=n$</span> for some integers <span class="math-container">$k,n$</span>. We have that <span class="math-container">$0 \leq x_1,...,x_k$</span> and individually <span class="math-container">$x_1 \leq a_1$</span>, <span class="math-container">$x_2 \leq a_2$</span>, .... | Ross Millikan | 1,827 | <p>There are six regions around the exterior that are not part of any of the smaller circles. You are asked what percentage of the large circle they represent. The inner area is divided into lenses and triangles with curved sides, which we can call deltas. You should compute the area of a lens and a delta, count how... |
3,427,794 | <p>Let's see I have the following equation</p>
<p><span class="math-container">$$
x=1
$$</span></p>
<p>I take the derivate of both sides with respect to <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$$
\frac{\partial }{\partial x} x = \frac{\partial }{\partial x}1
$$</span></p>
<p>The... | eyeballfrog | 395,748 | <p>If we're taking derivatives, that means the things on each side of the equals sign are functions. What you've demonstrated is the correct statement that if <span class="math-container">$f$</span> is the identity function <span class="math-container">$f(x) = x$</span> and <span class="math-container">$g$</span> is th... |
3,746,412 | <blockquote>
<p>Prove: For all sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, ( <span class="math-container">$A \cap B = A \cup B \implies A = B$</span> )</p>
</blockquote>
<p>In the upcoming proof, we make use of the next lemma.</p>
<p>Lemma: For all sets <span class="math-co... | halrankard | 688,699 | <p>As others have already noted, it is indeed much simpler to prove this by contrapositive. But you asked if <em>your</em> proof is correct, which is a perfectly reasonable question that I would like to answer.</p>
<p>There is a gap in your logic.</p>
<p>You have set out to prove:</p>
<p><span class="math-container">$$... |
2,421,344 | <p>I'm studying the weak Mordell-Weil theorem in Silverman's "Arithmetic of elliptic curves" and I can't understand the proof of proposition 1.6 of chapter 8.</p>
<p>Here $K$ is a number field, $S$ a finite set of places containing the archimedean ones and $m>2$ an integer. I want to prove that the maximal abelian ... | farruhota | 425,072 | <p>Alternatively:
$$
\begin{array}{c|l|c}
X (\# \ of \ bad) & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(X) & P(X) \\
\hline
0 & P(GGG)= \frac{7}{10}\cdot \frac{6}{9}\cdot \frac{5}{8} & \frac{210}{720} \\
1 & P(BGG)+P(GBG)+P(GGB) =\fra... |
603,469 | <p>Ok, so I'm not very good with these proving by induction thingies. Need a little help please.</p>
<p>How do I prove </p>
<p>$$\sum_{j=0}^n\left(-\frac12\right)^j=\frac{2^{n+1}+(-1)^n}{3\cdot2^n}$$</p>
<p>when $n$ is a non-negative integer.</p>
<p>I got the basis step and such down, but I'm pretty bad with expone... | Jean-Claude Arbaut | 43,608 | <p>Induction is a bit overkill here.</p>
<p>If $a \not = 1$,</p>
<p>$$\sum_{j=0}^n a^j=\frac{a^{n+1}-1}{a-1}$$</p>
<p>Here $a=-\frac 1 2$.</p>
<p>$$\frac{a^{n+1}-1}{a-1}=\frac{(-1/2)^{n+1}-1}{-1/2-1}=\frac{2}{3}\left(-\left(-\frac 1 2\right)^{n+1}+1\right)$$
$$=\frac{(-1)^n+2^{n+1}}{3 \cdot 2^n}$$</p>
|
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