Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Probability of number formed from dice rolls being multiple of 8
A fair 6-sided die is tossed 8 times. The sequence of 8 results is recorded to form an 8-digit number. For example if the tosses give {3, 5, 4, 2, 1, 1, 6, 5}, the resultant number is $35421165$. What is the probability that the number formed is a multip... | We only care about the last three digits obviously, so the possible sums of just those three die are $100a + 10b + c$, where $a$, $b$, and $c$ are all between $1$ and $6$, inclusive.
We check the divisibility by taking this modulo $8$. If the result is equivalent to $0$, then it is divisible.
Setting up the equivalence... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding a tangent to an ellipse parallel to a given line Problem: Find the lines that are tangent to the ellipse $x^2 + 4y^2 = 8$ and parallel to $x +2y = 6$.
I tried to find the derivative of $x^2 + 4y^2 = 8$ and I got: $$\frac{dx}{dy} = -\frac{x}{2y}.$$
Not quite sure if it's right, but I tried to equate it with the ... | Rewrite the equation of the ellipse $x^2+4y^2=8$ as
$$
\begin{align}
x^2+4y^2&=8\\
4y^2&=8-x^2\\
y^2&=\frac{8-x^2}{4}\\
y&=\left(\frac{8-x^2}{4}\right)^{\frac12}.\tag1
\end{align}
$$
The slope of the lines that are tangent to the ellipse is
$$
\begin{align}
\frac{dy}{dx}&=\frac{d}{dx}\left(\frac{8-x^2}{4}\right)^{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Generating Functions in Discrete Math a)Find the coefficient of $x^3y^4$ in $(2x + 5y)^7$. b) Find the coefficient of $x^5$ in $(3x -1)(2x +1)^8$.
I know this has to do with generating functions , but i'm not sure how to start with this problem in order to find the coefficient of part a) and b)
| For (b), you have $(3x - 1)(2x + 1)^{8}$. So consider $(2x+1)^{8}$. You are interested in the coefficients of $x^{4}$ and $x^{5}$ in that term. When you multiply by $(3x-1)$ you can form $x^{5}$ by $3x * kx^{4}$, and $x^{5}$ is held constant by multiplying with $-1$. So $(2x + 1)^{8}$ has coefficient of $x^{5}$ as $\bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/760978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Recurrence relation with generating function problem I've got a recurrence problem that I'm close to solving, but having trouble with finishing up.
Solve the following recurrence relation using generating functions:
$$g_n = g_{n-1} + g_{n-2} + n$$
for $ n>=2, g_0 = 1, g_1 =2 $.
What I've done so far:
$$G(x) = \s... | A simpler way to set this kind of problems up is to write the recurrence with no subtraction in indices:
$$
g_{n + 2} = g_{n + 1} + g_n + n + 2
$$
Multiply by $x^n$, sum over $n \ge 0$, and recognize:
\begin{align}
\sum_{n \ge 0} a_{n + k} x^k
&= \frac{G(x) - g_0 - g_1 z - \ldots - g_{k - 1} x^{k - 1}}{x^k} \\
\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/761378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Solving the non-homogeneous recurrence relation: $g_{n} = 12g_{n-2}-16g_{n-3}+6\cdot 2^n+25n$ $g_{n} = 12g_{n-2}-16g_{n-3}+6\cdot 2^n+25n$ With initial conditions $g_{0} = 23, g_{1} = 37, g_{2} = 42 $
This is a practice question I'm working on, and I'm running into absurd amounts of calculations with everything I have ... | Simplest way I know: Define the generating function:
$$
G(z) = \sum_{n \ge 0} g_n z^n
$$
Write the recurrence without subtraction in indices:
$$
g_{n + 3}
= 12 g_{n + 1} - 16 g_n + 48 \cdot 2^n + 25 n + 75
$$
If you multiply by $z^n$, sum over $n \ge 0$, and recognize the resulting sums:
\begin{align}
\sum_{n \ge 0} ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Which is larger $\sqrt[99]{99!}$ or $\sqrt[100]{100!}$ Which is larger
$\sqrt[99]{99!}$ or $\sqrt[100]{100!}$
I know that it is the $\sqrt[100]{100!}$
but is there a formula to figure this out instead of doing it all out by hand?
| Let $x=\sqrt[99]{99!}$ and $y=\sqrt[100]{100!}$, then
\begin{align}
\frac{x}{y}&=\frac{\sqrt[99]{99!}}{\sqrt[100]{100!}}\\
&=\frac{(99!)^\frac{1}{99}}{(100!)^\frac{1}{100}}\\
&=\frac{(99!)^{\frac{1}{100}+\frac{1}{9900}}}{(100!)^\frac{1}{100}}\\
&=\frac{(99!)^{\frac{1}{100}}(99!)^{\frac{1}{9900}}}{(100!)^\frac{1}{100}}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/763320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 1
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Solving $L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ priveded $a+b+c=0$
Let $a,b,c$ be such that $a+b+c=0$ and suppose that
$$L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}.$$
Find the value of $L$.
I can only see the symmetry of these function but cannot solve it.
| Since $a+b+c=0$ it follows $c=-a-b$, then
\begin{align}
2a^2+bc&=2a^2+b(-a-b)\\
&=2a^2-ab-b^2\\
&=(a-b)(2a+b)\\
&=(a-b)(a+a+b)\\
&=(a-b)(a-c)\quad\text{because $a+b=-c$}
\end{align}
In a similar way we can obtain $2b^2+ac=(b-a)(b-c)$ and $2c^2+ab=(c-a)(c-b)$. Then
\begin{align}
L&=\frac{a^2}{(a-b)(a-c)}+\frac{b^2}{(b-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/765440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find The Minimum Value of the quantity Find the minimum value of the quantity $$\frac{(a^2+3a+1)(b^2+3b+1)(c^2+3c+1)}{abc}$$,where $$a,b,c>0$$ and $$ a,b,c\in R $$are positive real numbers.
| $\dfrac{a^2 + 3a + 1}{a} = a + \dfrac{1}{a} + 3 \geq 2 + 3 = 5$ by AM-GM. So $LHS \geq 5^3 = 125$ with equality at $ a = b = c = 1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}$? find the integral of $f(x)=\frac1{(x-1)^2\sqrt{x^2+6x}}$
my attempt =
$(x-1)=a$, $a=x+1$ so the integral'd be
$\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}=\int\frac{da}{a^2\sqrt{a^2+8a+7}} $
lets say $\sqrt{a^2+8a+7}=(a+1)t$
so $a=\frac{7-t}{t-1}$ and $da=\frac{-6dt}... | Let $\displaystyle a=\frac{7-t^2}{t^2-1}$ instead of $\displaystyle a=\frac{7-t}{t-1} $ ??
Doing that and simplifying we get, $\displaystyle -\frac{12 t}{\left(t^2-1\right)^2}$ and integrating, we get
$$-2 \left(-\frac{3 t}{7 \left(t^2-7\right)}+\frac{2 \log \left(\sqrt{7}-t\right)}{7 \sqrt{7}}-\frac{2 \log \left(t+\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/768020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Solve $\sin x+\cos x+\tan x+\csc x+\sec x+\cot x=-2$ in the interval $0
Solve $$\sin x+\cos x+\tan x+\csc x+\sec x+\cot x=-2$$ from $0<x<2\pi$.
Could you find the most elegant solution? Does it factorize?
| This might be considered an extended comment of lab bhattacharjee's answer above.
The roots of the factored form are
$(\cos x + \sin x)(\cos x + 1)(\sin x +1) = 0$
which gives
$\cos x = -\sin x$. The 2 solutions are $\dfrac{3\pi}{4}$ and $\dfrac{7\pi}{4}$;
$\cos x = -1$. The solution is $\pi$;
$\sin x = -1$. The solu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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If n+1 is a cube show that 504 divides n(n+1)(n+2) If $n+1$ is a cube show that 504 divides $n(n+1)(n+2)..$
I try that : $n+1=a^3$ , then $n=a^3-1$ so $n(n+1)(n+2) = (a^3-1)a^3(a^3+1) = (a^6-1)a^3$ and I try prove $(a^6-1)a^3$ divides by 504 with induction but it is not easy (or short). I just find anybody who solve t... | You need to show that $(a^3-1)a^3(a^3+1)$ is divisible by $9$, $8$, and $7$ (since $504=9\cdot8\cdot7$). Your idea of rewriting the product at $(a^6-1)a^3$ is a good one, since that takes care of $7$, using Fermat's little theorem.
For the other two, you can factor the product at $k^3(k-1)(k+1)(k^2+k+1)(k^2-k+1)$. Fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/772949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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finding a basis for kernal and image when dealing with a matrix vectorspace
I've calculated the dimension of $M_2(\mathbb{F})$ as $4$, and calculated $T(X) = \begin{pmatrix} -(x_2+x_3) & x_1 - x_4 \\ x_1 - x_4 & x_3 + x_2 \end{pmatrix}$ and noted that $T(X) = 0$ iff $x_2 = -x_3$ and $x_1 = x_4$ but I'm not sure how I ... | Consider arbitrary element
$$
X =\begin{pmatrix} x_1 & x_2 \\ x_3 & x_4 \end{pmatrix}\in \ker T
$$
As you have calculated, $x_2 = -x_3$, $x_1=x_4$, thus
$$
X =\begin{pmatrix} x_1 & -x_3 \\ x_3 & x_1 \end{pmatrix}=
\begin{pmatrix} x_1 & 0 \\ 0 & x_1 \end{pmatrix}+
\begin{pmatrix} 0 & -x_3 \\ x_3 & 0 \end{pmatrix}=
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/774825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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If $\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$ then $x^2 - 44x - 36 = 0$ holds for $x=\sin 2A$ If
$$\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$$
then prove that
$$\sin 2A \quad\text{ is a root of }\quad x^2 - 44x - 36 = 0$$
I have no idea how to solve it. Plz help.
| Let $$\sin{A}+\cos{A}=t$$
$$7=\sin A+\cos A+\frac{1}{\sin A\cos A}+\frac{\sin A+\cos A} {\sin A\cos A}$$
$$\sin 2A =2 \sin A \cos A$$
$$t^2=1+\sin 2A$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/775269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluate limits by interpreting sums as integral sums Problem: Evaluate the following limits by interpreting given sums as integral sums for certain functions and by using the Fundamental Theorem of Calculus. (a) Find $\lim{S_{n}}$ as n goes to infinity where $$S_{n} = \frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{2n}... | We can rewrite our sum $S_n$ as
$$S_n=\frac{1}{n}\left(\frac{1}{1+\frac{1}{n}}+ \frac{1}{1+\frac{2}{n}}+\frac{1}{1+\frac{3}{n}}+\cdots +\frac{1}{1+\frac{n}{n}}\right).$$
Note that $\dfrac{1}{1+\frac{k}{n}}$ is the value of the function $f(x)=\frac{1}{1+x}$ at the point $x=\frac{k}{n}$.
So we have evaluated the functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/778435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Combinatorial Problem From Book Find all the number of ways such that three elements selected from the set $\{ 1,2,3,....,4n\}$ sum of whose is divisible by $4$.
I cannot find the solution of this equation $x + y + z = 4k$. Solving this equation we can get our answer.
| We can do the job by considering various cases. There are various ways in which the sum can be divisible by $4$.
Case (i): The $3$ chosen numbers are all congruent modulo $4$ (have the same remainder on division by $4$). Then they must all be congruent to $0$ (have remainder $0$). Since $n$ of our numbers are divis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares. I have figured out a (long, and tedious) way to do it. But I was wondering if there is some sort of direct correlation or another path that I completely missed.
My attempt at the program was as follows:
A number of the form, $8k +... | You get a little bonus: $x^2 + y^2 + z^2$ can be even with two of the variables odd. However, $x^2 + y^2 + z^2$ cannot be divisible by $4$ unless all three of $x,y,z$ are even. So, Assume $x^2 + y^2 + z^2 \equiv 28 \pmod {32}.$ It follows that $x,y,z$ are even, and we get $(x/2)^2 + (y/2)^2 + (z/2)^2 \equiv 7 \pmod 8.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Probability of rolling 3 sixes in 6 rolls I roll a die six times. What is the probability of rolling at least 3 sixes? Please share the formula as well, so I can figure it out myself in future.
| Let $p$ = probability of getting a $6$ for each toss, then $p = \dfrac{1}{6}$, and $q = 1 - p = \dfrac{5}{6}$ is the probability of getting a number other than $6$. Let $k$ = the number of successful trials. This is a binomial experiment consisting of $6$ independent trials. So $n = 6$. Let $x$ be the number of $6$'s i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$
Here, $AM_1$ is the angle bisector of $\angle A$ extended to the circumcircle and so on. $R$ is the circumradius and $r$ is the inradius, respectively. I have to prove that:
$$8r+2R\le AM_1+BM_2+CM_3\le 6R$$
The second part is easy, since each of $AM_1$ i... | The quadrilaretal $ABM_1C$ is cyclic. Thus the Ptolemy Theorem says:
$AB.CM_1+AC.BM_1=BC.AM_1$, that is $AM_1=\dfrac{b+c}{a}.BM_1$
(Since, angle subtended by equal chords at the circumcircle are equal and vice-versa, $BM_1=CM_1$, as $\angle BAM_1=\angle CAM_1=\dfrac{A}{2}$).
Also, $BM_1=2R\sin\frac{A}{2}$.
Similarly, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Last Two digits of ${14}^{{14}^{14}}$ How to calculate the last two digits of ${14}^{{14}^{14}}$? With the help of any method. I have tried and have got the last digit to be $6$. But not sure.
| Clearly, $14^{14^{14}}$ is a multiple of $4$. To compute $14^n\pmod{25}$ we should know $n\pmod {\phi(25)}$, i.e. $14^{14}\pmod{20}$. Again, $14^{14}$ is a multiple of $4$, and it is $\equiv (-1)^{14}\equiv 1\pmod 5$. Hence $14^{14}\equiv 16\pmod {20}$. Thus $14^{14^{14}}\equiv 14^{16}\pmod {25}$. This can me computed ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/783467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculus Question: $\int_0^1\frac{(x-x^2)^4}{1+x^2}dx$ I have no idea how to find this integral:
$$\int_0^1\frac{(x-x^2)^4}{1+x^2}\ dx\ ?$$
Any help would be appreciated. Thanks in advance.
| HINT :
$$
\begin{align}
\int_0^1\frac{(x-x^2)^4}{1+x^2}\ dx&=\int_0^1\frac{x^4-4x^5+6x^6-4x^7+x^8}{1+x^2}\ dx\tag1\\
&=\int_0^1\left(x^6-4x^5+5x^4-4x^2+4-\frac4{1+x^2}\right)\ dx.\tag2\\
\end{align}
$$
See binomial theorem for $(1)$ and polynomial long division for $(2)$. Also
$$
\begin{align}
\int_0^1\frac1{1+x^2}\ dx... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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Intersection multiplicity
Let $f=y^2-x^3$ and $g=y^3-x^7$. Calculate the intersection multiplicity of $f$ and $g$ at $(0,0)$.
I know the general technique for this (passing to the local ring) but I having difficulty with the fact that $3,7,2$ have no common factors.
| I think you no longer need an answer, but I’d like to write something here since it may help some other students.
We need four lemmas here(You can find them in lecture 2 of Andreas Gathmann’s notes in Plane Algebraic Curves):
Lemma (1): For any three plane curves F, G, H, $$\mu_p(F, G)=\mu_p(F, G+FH)$$
Lemma (2): For a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Find all real $x$ ,such $8x^3-20$ and $2x^5-2$ is perfect square
Find all real numbers $x$,such
$$8x^3-20,2x^5-2$$ is the perfect square of an integer
My idea: First we find the real number $x$ such $$8x^3-20,2x^5-2$$ is postive integer numbers,and second the real numbers such
$$8x^3-20=m^2,2x^5-2=n^5$$
How prov... | Hint. If we have
$$8x^3-20=m^2\ ,\quad 2x^5-2=n^2$$
then
$$x^3=\frac{m^2+20}{8}\ ,\quad x^5=\frac{n^2+2}{2}\ .$$
Can you see how to obtain $x$ from these two equations?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/784633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Indefinite Integral $\int \frac{dx}{\sqrt {ax^4-bx^2}}$ I am trying to Integrate
$$
I=\int \frac{dx}{\sqrt {ax^4-bx^2}}, \qquad a,b\in \mathbb{R}.
$$
Thanks.
I tried to do $x=\sin \phi$
$$
\int \frac{\cos \phi\, d\phi}{\sqrt{a\sin^4 \phi-b\sin^2 \phi}}=\int \frac{\cot \phi \, b\phi}{\sqrt{a\sin^2\phi-b}}
$$
but get stu... | $$
\begin{aligned}
\int\frac{\mathrm{d}x}{\sqrt{ax^4 - bx^2}}&=\int\frac{\mathrm{d}x}{\sqrt{bx^4\left(\frac{a}{b} - \frac{1}{x^2}\right)}}\\
&=\frac{1}{\sqrt{b}}\int\frac{1}{\sqrt{\frac{a}{b} - \left(\frac{1}{x}\right)^2}}\frac{1}{x^2}\,\mathrm{d}x
\end{aligned}
$$
Now, set $u=\dfrac{1}{x}$ and $\mathrm{d}u=-\dfrac{1}{... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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block matrix multiplication If $A,B$ are $2 \times 2$ matrices of real or complex numbers, then
$$AB = \left[
\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot
\left[
\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right]
=
\left[
\begin{array}{cc} a_{11}b_{11}+a_{12}b_{21... | It depends on how you partition it, not all partitions work.
For example, if you partition these two matrices
$$\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix},
\begin{bmatrix}
a' & b' & c' \\
d' & e' & f' \\
g' & h' & i'
\end{bmatrix} $$
in this way
$$ \left[\begin{ar... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $a+b+c+d$ If $a$, $b$, $c$, and $d$ are distinct integers such that
$$(x-a)(x-b)(x-c)(x-d)=4$$
has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$?
I tried to analyze graphically by shifting the graph of
$f(x)=(x-a)(x-b)(x-c)(x-d)$ four units downward but couldn't infer anything due t... | The product of the four distinct integers $r-a$, $r-b$, $r-c$, and $r-d$ is $4$. Thus $r-a,r-b,r-c,r-d$ are, in some order, $-1$, $1$, $-2$, and $2$.
We may assume it is in the order we gave. So $r-a=-1$, $r-b=1$, $r-c=-2$, $r+d=2$. Solve for $a,b,c,d$ and add.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/787979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Non-unitary representation How to prove $\pi :\mathbb R\to \mathbb C^2$, defined by $t\mapsto \begin{pmatrix} 1 & t\\ 0 & 1\end{pmatrix}$ is a non-unitary representation?
Is the following correct? $\pi$ is a representation because
$$\pi (t+s)=\begin{pmatrix}1 & t+s \\ 0 & 1\end{pmatrix}=\begin{pmatrix}1 & t\\ 0 ... | For $\pi$ to be a unitary representation, $\pi(x)$ must be a unitary matrix for every element $x$ of the group. Here,
$$\begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix}$$
is not a unitary matrix unless $t=0$. Indeed,
$$\begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix}^* \begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix}
=\begin{pmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/788743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$ using complex analysis. Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$. This is the last question in our review for complex analysis. Hints were available upon request, but being the student I am, I waited until the last minute to do this. Can anyone give ... | Consider the contour integral
$$\oint_C dz \frac{\log^3{z}}{z^2+4} $$
where $C$ is a keyhole contour about the positive real axis, with outer radius $R$ and inner radius $\epsilon$. The contour integral is then equal to
$$\int_{\epsilon}^R dx \frac{\log^3{x}}{x^2+4} +i R \int_0^{2 \pi} d\theta \, e^{i \theta} \frac{\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
Area calculation of ellipse $x^2/2+y^2=1$ Calculate the area of the ellipse that you get when you rotate the ellipse $$\frac{x^2}{2}+y^2= 1$$ around the x-axis.
My approach has been to use the formula for rotation area from $-2$ to $2$. But this gives a complicated integral and I'm unsure about the limits $-2$ to $2$.
... | Surface
$\frac{x^2}{2}+y^2=1$
$S=2\pi\int_{-a}^{a}yds=4\pi\int_{0}^{a}yds$
$y'= -\frac{x}{2y}$
$ds=\sqrt{1+y'^2}dx=\sqrt{1+\frac{x^2}{4y^2}}dx=\frac{\sqrt{4y^2+x^2}}{2y}dx$
$S=4\pi\int_{0}^{\sqrt{2}}y\frac{\sqrt{4y^2+x^2}}{2y}dx=2\pi\int_{0}^{\sqrt{2}}\sqrt{4y^2+x^2}dx=2\pi\int_{0}^{\sqrt{2}}\sqrt{4-x^2}dx$
Substituti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/793036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
no. and nature of roots of $x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$ The given equation is
$$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$
I took $\log_{2}{x}$ = $t$
and then rewrote the given equation as
$$x^{3t^2 + 4t - 5} = \sqrt{2}$$
But I don't know what to... | $$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$
$$\log_2 {x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}}} = \log_2 {\sqrt{2}}$$
$$\log_{2}{x} ({\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}})=\frac12$$
$t=\log_2{x}$
$$3t^3+4t^2-5t-2=0$$
$t_1=1.$
Can you finish?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/794231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show $(n-1)^3n^3(n+1)^3$ is divisible by 7 and 9? Yeah it looks like a basic, really elementary question, but i'm having hard time with it.
First i tried to show that it's divisible by 9
$$(n-1)^3n^3(n+1)^3 = ((n+1)(n-1))^3n^3 = (n^2-1)^3n^3 = (n^3-n)^3$$
and using eulers theorem we know that
$$[n^{\varphi(9)} ... | It's divisible by $9$ and in fact $27$, but not necessarily by $7$.
To see it is divisible by $27$, use the fact that that one of $n$, $n - 1$, $n - 2$ is divisible by $3$.
Your way
Expand out what you have:
$$
(n^3 - n)^3
= n^{9} - 3 n^{7} + 3n^5 - n^3
$$
If $n$ is divisible by $3$, you are done.
Otherwise, as you not... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/794478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Method of characteristics for a system of pdes
I can do parts a) and b) as follows
$\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1\end{pmatrix}\frac{\partial}{\partial{}x}\begin{pmatrix} u \\ v \\ w\end{pmatrix}+\begin{pmatrix} 1&1&0 \\ 1&2&1 \\ 0&1&1\end{pmatrix}\frac{\partial}{\partial y}\begin{pmatrix} u \\ v \\ w\end{pma... | The solution is done in two steps.
First, diagonalise the linear system. We denote by $\mathbf{U}=[u;v;w].$ The original system becomes
$$\partial_x\mathbf{U}+M\partial_y\mathbf{U}=0,$$ where
$$
M=\begin{pmatrix}
1 & 1 & 0\\
1 & 2 & 1\\
0 & 1 & 1
\end{pmatrix}.
$$
We first diagonalise $M.$ We have (using Wolfram Alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/799360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Find the area of the trapezium ABCD is trapezium AB||CD. 10 & 40 are the areas of the respective parts
How to find out the area of the trapezium?
| $$\triangle COD \sim \triangle AOB.$$
If $\dfrac{CD}{AB}=a$, then $\dfrac{\text{Area of}\space \triangle COD}{\text{Area of} \space \triangle AOB}=a^2$.
So, we conclude: $\dfrac{CD}{AB}=\sqrt{\frac{40}{10}}=\sqrt{4}=2$.
If we denote trapezium total height $h$, then
heights of corresponding triangles are $\frac{2h}{3}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For which $n$ is $ \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin nx}= \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin x}=\frac{\pi}{3\sqrt{3\,}\,}$? I have been trying to figure out for which $n$ is $$ \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin nx} = \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin x}=\frac{\pi}{3\sqrt{3\,}\,}$$
Using maple... | Although the problem has been solved a different method is presented here.
The problem asks to find which values of $n$ that satisfy
\begin{align}
\int_{0}^{\pi/2} \frac{dx}{2 + \sin(nx)} = \int_{0}^{\pi/2} \frac{dx}{2 + \sin(x)}
= \frac{\pi}{3 \sqrt{3}}.
\end{align}
The solution proposed is as follows.
Consider the i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 2
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Binomial Sum Related to Fibonacci: $\sum\binom{n-i}j\binom{n-j}i=F_{2n+1}$ How would I prove
$$
\sum\limits_{\vphantom{\large A}i\,,\,j\ \geq\ 0}{n-i \choose j} {n-j \choose i}
=F_{2n+1}
$$
where $n$ is a nonnegative integer and $\{F_n\}_{n\ge 0}$ is a sequence of Fibonacci numbers?
Thank you very much! :)
| The problem asks to show that
\begin{align}
\sum_{i=0}^{n} \sum_{j=0}^{n} \binom{n-i}{j} \binom{n-j}{i} = F_{2n+1}.
\end{align}
The problem, as stated, is incorrect. It should read $F_{2n+2}$. This will be shown
in the following.
Consider the double summation
\begin{align}
S_{n} = \sum_{i=0}^{n} \sum_{j=0}^{n} \binom{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 0
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derivative of $y=(x^2+x^3)^4$ I can't figure out where I am going wrong.
$$y=(x^2+x^3)^4$$
chain rule it first
$$4(x^2+x^3)^3* \frac d{dx}(x^2+x^3)$$
which should become:
$$4(x^2+x^3)^3(2x+3x^2)$$
factoring out should give me:
$$4*x^2*x(1+x)^3(2+3x)$$
which to me says the answer is: $4x^3(1+x)^3(2+3x)$
but the book sa... | Your mistake is in the factoring out. When you factor $x^2$ out from the $(x^2+x^3)^3$ term, it becomes $(x^2)^3(1+x)^3=x^6(1+x)^3$ instead of $x^2(1+x)^3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/801897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are the steps to derive the following inverse Fourier transformations I'm reading a text which is an introductory text on Fourier transforms. The author has two expressions:
$$ F(\omega_{o}) = \frac{1}{\sigma \sqrt {2 \pi} } e^{\Large- \frac{{\omega_{o}}^2}{2\sigma^2}} $$
and
$$G(\omega_{o}) = 2\pi \cos(\omega_{o... | For the first one, we have
$$
F(\omega)=\frac{1}{\sigma\sqrt{2\pi}} e^{\Large- \frac{\omega^2}{2\sigma^2}}.
$$
It seems that you are using the textbook for physics or engineering, so the inverse Fourier transform of $F(\omega)$ using the 'convention' notation in those fields is
$$
\begin{align}
\mathcal{F}^{-1}[F(\omeg... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Proof for $\sin(x) > x - \frac{x^3}{3!}$ They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried:
$\sin(x) + x -\frac{x^3}{6} > 0 \\$
then I computed the derivative of that function to d... | A trick is to get rid of the transcendental function.
As the sine is monotonic in $(0,\pi/2)$,
$$x>\arcsin\left(x-\frac{x^3}3\right),$$ and, as the two members coincide for $x=0$, by differentiation
$$1>\frac{1-x^2}{\sqrt{1-\left(x-\dfrac{x^3}3\right)^2}}.$$
Then it suffices that for $x<1$,
$$1-\left(x-\dfrac{x^3}3\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 9,
"answer_id": 7
} |
Inequality involving conjugate numerator/denominator pairs Question is to solve:
$$\frac{(x-2)(x-4)(x-7)}{(x+2)(x+4)(x+7)} > 1$$
I thought I could negate terms to make them equal (i.e. $-(x-2)$), but that does not happen. I could subtract $1$ from each side but that would be $(very)^2$ heavy work. Is there any simple... | Another way would be to multiply throughout by $(x+2)^2(x+4)^2(x+7)^2$, which is positive ($x =-2,-4,-7$ are not allowable anyway), to get
$$(x^2-4)(x^2-16)(x^2-49)>(x+2)^2(x+4)^2(x+7)^2$$
Collecting terms in the LHS and factoring, this is equivalent to
$$-2(x+7)(x+4)(x+2)(13x^2+56)>0$$
and now checking intervals whe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For which real numbers $c$ is $\frac{e^x+e^{-x}}{2} \le e^{cx^2}$ for all real numbers $x$? This question comes from the 1980 Putnam exam. My work is shown below.
For all integers $n \ge 1$, \begin{align}
(2n)!&=n!\cdot\underbrace{(n+1)(n+2)(n+3)\cdots(2n-2)(2n-1)(2n)}_{n \text{ terms}} \\
&\ge n! \cdot \underbrace{2... | The method presented in the proposal of the problem is correct.
An alternate method is:
\begin{align}
\cosh(x) \approx 1 + \frac{x^{2}}{2} + \mathcal{O}(x^{4})
\end{align}
and
\begin{align}
e^{c x^{2}} \approx 1 + c x^{2} + \mathcal{O}(x^{4})
\end{align}
for which the inequality
\begin{align}
e^{c x^{2}} \geq \cosh(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 0
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Number of Rolls of Fair Dice to get '6' and '5' A Fair Dice is Thrown Repeatedly. Let $X$ be number of Throws required to get a '$6$' and $Y$ be number of throws required to get a '$5$'. Find $$E(X|Y=5)$$
| We cannot get both 5 and 6 on the fifth trial, so $P(X=5 \mid Y=5)=0$
$\mathrm{\large E}(X\mid Y=5) = \sum\limits_{x=1}^{4} x\cdot\mathrm{\large P}(X=x\mid Y=5) + \sum\limits_{x=6}^{\infty} x\cdot\mathrm{\large P}(X=x\mid Y=5)$
To unconditionally get a 5 on the fifth trial, we need four not 5 before a 5 on the fifth.$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Series Question: $\sum_{n=1}^{\infty}\frac{n^2}{(4n^2-1)^3}$ How to compute the following series:
$$\sum_{n=1}^{\infty}\frac{n^2}{(4n^2-1)^3}$$
I tried to use partial fraction
$$\begin{align}\frac{n^2}{(4n^2-1)^3}&=\frac{1}{64(2n+1)}-\frac{1}{64(2n-1)}+\frac{1}{64(2n+1)^2}+\frac{1}{64(2n-1)^2}\\&-\frac{1}{32(2n+1)^3}+\... | Note that:
$$\sum _{n=1}^{\infty }{\frac {{n}^{2}}{ \left( 4\,{n}^{2}-1 \right) ^{3
}}}=\frac{1}{4}\,\sum _{n=1}^{\infty }\frac{1}{\left( 4\,{n}^{2}-1 \right)^3 }+
\frac{1}{\left( 4\,{n}^{2}-1 \right)^2 }\tag{1}$$
then note that:
$$\sum _{n=1}^{\infty } \frac{1}{\left( {n}^{2}-{x}^{2} \right)}=\frac{1}{2x^2}-\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$?
How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$ ?
The numerator is a irreducible polynomial so I can't use partial fractions.
I tried the substitutions $t = x^2, t=x^4$ and for the formula $\int u\,dv = uv - \int v\,du$ I tried using: $u=\frac{x^4 + 1 }{x^6 + 1} , \,dv=\,d... | OK it is a rational function so we preform a partial fraction decomposition.
The first step is to factor the denominator into linear and quadratic factors.
$$x^6+1=(x^2+1)(x^4-x^2+1)=(x^2+1)((x+1)^2-x^2))=
(x^2+1)(x^2-x+1)(x^2+x+1)$$
Now we find
$$\frac{x^4+1}{x^6+1}=\frac{2}{3}\frac{1}{x^2+1}+\frac{1}{6}\frac{1}{x^2-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/811911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Integrals of a particular form Recently, I was given a question sheet with a lot of integrals, and I could solve all of them except for a particular type of them:
$$I_1=\int \frac{dx}{(x+1)^2\sqrt{x^2+2x+2}}=\int \frac{dx}{x^2\sqrt{x^2+1}}=\frac{-\sqrt{x^2+1}}{x}+C$$
$$I_2=\int \frac{dx}{(x-1)^2\sqrt{x^2-x+1}}$$
$$I_3=... | Usually more direct, with $\sqrt {u^2 + 1},$ to take $u = \sinh t.$ With $\sqrt {u^2 - 1},$ to take $u = \cosh t.$ Tends to save several steps.
Relevant: it will still be necessary, to integrate a rational function of $\sinh x$ and $\cosh x,$ to use a stereographic projection. I worked it out:
$$ v = \tanh \frac{x}{2}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing that $3x^2+2x\sin(x) + x^2\cos(x) > 0$ for all $x\neq 0$ I got this question:
Show that for all $x\neq 0$, $3x^2+2x\sin(x) + x^2\cos(x) > 0$
I tried to show it but got stuck.
| If $x<0$ then
$$\sin x<-x\implies 2x\sin x>-2x^2$$
and
$$\cos x\ge-1\implies x^2\cos x\ge-x^2$$
and so
$$3x^2+2x\sin x+x^2\cos x>3x^2-2x^2-x^2=0\ .$$
The case $x>0$ is similar, even a bit easier.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/812453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$
Compute the following integral
\begin{equation}
\int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, \exp\left[\frac{x^2-1}{x^2+1}\right]... | Rewrite
\begin{align}
&\int\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\ \exp\left[\frac{x^2-1}{x^2+1}\right]\ dx\\
&=-\int\left[\frac{(1-x^2)\ln\left(\dfrac{1-x^2}{1+x^2}\right)-(1+x^2)}{(1-x^2)(1+x^2)(1+x^2)}\right] x\ \exp\left[-\frac{1-x^2}{1+x^2}\right]\ dx\\
&=-\frac14\int\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/815863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "116",
"answer_count": 2,
"answer_id": 1
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Creating two groups where two people can't be in the same group. If you have a group of $12$ men and $10$ women, and you need to make two groups - one with $6$ people and the other with $9$, the ways to form such groups would be
$${22 \choose 6}\cdot {16 \choose 9}$$
Since I guess the order doesn't matter.
But then, th... | In the initial case, you do not distinguish between men and women. So effectively, you have $22$ distinct people to make up two groups one of which has $6$ members and the other has $9$ members. Let's assume we have assembled all possible groups of $6$. For each group of $6$, there would be $n$ sets of 9 elements fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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A little Problem in Trigonometry (Multiple Angle) If $\tan^2 \theta = 1 + 2\tan^2 \phi$, show that $\cos 2\phi = 1 + 2\cos2\theta$.
What I have done..
$$\implies \tan^2 \theta = 1 + 2\tan^2 \phi\\
\implies 1 + \tan^2 \theta = 2 + 2\tan^2 \phi\\
\implies 1 + \tan^2 \theta = 2(1 + \tan^2 \phi)\\
\implies \sec^2 \theta = ... | Your steps are fine you just need to do the following (I'm putting your steps first):
\begin{align*}
\tan^{2}\theta & = 1+2 \tan^{2}\phi\\
1+\tan^{2}\theta & = 2+2 \tan^{2}\phi\\
\sec^2 \theta & = 2 \sec^2 \phi\\
\text{Do the following to continue:}\\
\frac{1}{\cos^2 \theta} & = \frac{2}{\cos^2 \phi}\\
\cos^2 \phi & =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/817521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}
Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded.
Show that there exits a positive integer $k$ such that, when $n>k$
$$\dfrac{a_{1}}{a_{2}}+\dfrac{a_{2}}{a_{3}}+\cdots... | Let $(1-\epsilon_n) = a_{n} / a_{n+1}$, Then $a_n = a_1 / \prod_{i=1}^{n-1} (1-\epsilon_i)$. Since $a_n$ is unbounded, it follows that $\prod_{i=1}^\infty (1-\epsilon_i) = 0$. It is well known that this product equals zero if and only if $\sum_{i=1}^\infty \epsilon_i$ diverges.
But $\frac{a_1}{a_2} + \cdots \frac{a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Simple algebra formula for which I can't find the right answer I have the formula $y + (z + 1) = \frac{1}{2} \cdot (z + 1) \cdot (z + 2)$, and I should work to $y = \frac{1}{2}\cdot z \cdot (z + 1)$.
Somebody showed me how it's done:
$y + (z + 1) = \frac{1}{2} \cdot (z + 1) \cdot (z + 2)$
$y + (z + 1) = \frac{1}{2} \cd... | In your step 4, you forgot to divide (3z+2) by 2
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/818475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Primes as a difference of powers Find the smallest prime that cannot be written as
$$|3^a - 2^b|$$
EDIT: I forgot to mention that $a$ and $b$ are whole numbers.
I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but I couldn't infer much. Please help. Thanks in advance!
| $41$ is the answer.
For $41=3^a-2^b$, clearly $b \not =0, 1, a \not =0$. $\pmod{3}$ gives $b$ even, $\pmod{4}$ gives $a$ even. Then $41=(3^{\frac{a}{2}}-2^{\frac{b}{2}})(3^{\frac{a}{2}}+2^{\frac{b}{2}})$, so $3^{\frac{a}{2}}-2^{\frac{b}{2}}=1, 3^{\frac{a}{2}}+2^{\frac{b}{2}}=41$, which is clearly impossible.
For $41=2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Stuck on finding the value of $\sum_{n=0}^\infty {n(n+1) \over 3^n}$ I am trying to find the value of the series
$$
\sum_{n=0}^\infty {n(n+1) \over 3^n}
$$
Here's what I have done so far:
$$
\sum_{n=0}^\infty {n(n+1) \over 3^n}=\sum_{n=0}^\infty {n^2 \over 3^n}+\sum_{n=0}^\infty {n \over 3^n}
$$
Determining the values... | Write it as:
$$
\frac{1}{3} \sum_{n \ge 0} (n + 2) (n + 1) \left( \frac{1}{3} \right)^n
$$
This looks suspicious...
$$
\frac{\mathrm{d}^2}{\mathrm{d} z^2} \frac{1}{1 - z}
= \sum_{n \ge 0} (n + 2) (n + 1) z^n
$$
and you are all set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/822059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Formula of signed distance from hyperplane to point Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$?
I found, that following value is the orientation (i.e. its sign matches ... | Finally, I found the solution (in book Andrew J. Hanson: Geometry for N-Dimensional Graphics):
$
\displaystyle
V=\frac{1}{n!}
\left\vert
\begin{array}{ccccc}
p_{1,1} & p_{1,2} & \cdots & p_{1,n} & 1 \\
p_{2,1} & p_{2,2} & \cdots & p_{2,n} & 1 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
p_{n,1} & p_{n,2} & \cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate $\int_{0}^{2\pi}{\sin^8x}\ {dx}$ I have started doing integration by parts:
$$\int_{0}^{2\pi}{\sin^8(x)}{dx}
= \int_{0}^{2\pi}{\sin^7(x)}\cdot{\sin(x)dx}
= \int_{0}^{2\pi}{\sin^7(x)}\cdot{d(-\cos(x))}
= \left. -\cos(x) \cdot \sin^7(x) \right|_0^{2\pi}
+ \int_{0}^{2\pi}{\cos(x)}{d(\sin^7(x))}
= \left. -... | $$
\begin{align}
\int_0^{2\pi}\sin^8xdx
&=\int_0^{2\pi}\left(\frac{e^{ix}-e^{-ix}}{2i}\right)^8dx\\
&=\int_0^{2\pi}2^{-8}(e^{ix}-e^{-ix})^8dx\\
&=2^{-8}\int_0^{2\pi}\sum_{k=0}^8\binom{8}{k}(e^{ix})^k(-e^{-ix})^{8-k}dx\\\
&=2^{-8}\sum_{k=0}^8\binom{8}{k}(-1)^{8-k}\int_0^{2\pi}e^{i(2k-8)x}dx\\
&=2^{-8}\sum_{k=0}^8\binom{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$
I tried to solve it in two ways and got a little stuck:
One way is to use Cauchy's MVT, define $f,g$ such that $f(x)=\sin x -x +\frac {x^3}{3!}$ and $g(x)=\frac ... | Observe first that $f_0(x):=1-\cos x>0$ for $0<x\leqslant\pi/2$. Then
$$f_1(x):=\int_0^xf_0(t)\,\mathrm dt=x-\sin x>0$$
for $0<x\leqslant\pi/2$. Continue in this way, with
$$f_{n+1}(x):=\int_0^xf_n(t)\,\mathrm dt>0$$
for $0<x\leqslant\pi/2$ and $n=0,1,...$, and we arrive at your result at the stage $n=4$.
It is easy t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/826015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Need algebra tip about $a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$ for sides of a triangle I just got a long expression:
$$a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$$
and I need to prove its less than zero for every $a$, $b$, and $c$ which are triangle sides
I really need tips how to handle such large expressio... | Others have mentioned it came from Heron's formula. That expresses the area of a triangle as a function of the lengths $a$, $b$, and $c$ of the three sides. Notice that since "a straight line is the shortest distance between two points", we must have $a+b\ge c$, $b+c\ge a$, and $c+a\ge b$. If $a+b$ happens to be equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/827304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
How many 10-bit strings with more 0’s than 1’s? I have to pick the answer from:
a.512
b.386
c.256
d.252
e.none of these
The number of bit strings of length 10 with n 0's (or n 1's in fact):
is C(10,n) , where C(a,b) = a! / [(a-b)!b!] is the combinitorial function.
So the first answer is C(10,3) = 120
In the second, the... | Exactly half of the strings that don't have $5$ of each. How many have $5$ of each? $\binom{10}{5}=252$ how many are there in total? $2^{10}=1024$.
Therefore the answer is $\frac{1024-252}{2}=386$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/828203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability depending on x being the difference between two numbers An urn contains 10 balls, labeled 1 – 10 . A set of two balls is drawn from the urn, and let X record the absolute value of the difference. Compute Prob( X < 2 ).
My understanding is that this is without replacement and so the probability of ... | Let us draw the balls, and then look at them one at a time.
There are two possibilities: (i) The first ball is $1$ or $10$ or (ii) The first ball is $2$ to $9$.
In Case (i), the probability that the second ball is a neighbour is $\frac{1}{9}$. In Case (ii), the probability the second ball is a neighbour is $\frac{2}{9}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $ I have been having extreme difficulties with this integral. I would appreciate any and all help.
$$
\int \sqrt{\tan x} ~ \mathrm{d}{x}.
$$
| As already mentioned in some answers, let $t^2=\tan x \implies 2tdt=\sec^2x dx\implies dx=\frac{2tdt}{t^4+1}$. Now, We can easily reach to the final answer as follows $$I=\int \frac{2t^2 dt}{t^4+1}=\int \frac{2 dt}{t^2+\frac{1}{t^2}}=\int \frac{\left(1+\frac{1}{t^2}\right)+\left(1-\frac{1}{t^2}\right) dt}{t^2+\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "61",
"answer_count": 7,
"answer_id": 3
} |
Let $f(x)$ denote the sum of the infinite trigonometric series, $f(x) =\sum^{\infty}_{n=1} \sin\frac{2x}{3^n}\sin\frac{x}{3^n}$ Let $f(x)$ denote the sum of the infinite trigonometric series,
$$f(x) =\sum^{\infty}_{n=1} \sin\frac{2x}{3^n} \sin\frac{x}{3^n}$$
I am not getting any clue how to solve it only thing I can p... | Hints:
$$\sin\alpha\sin\beta=\frac12\left(\cos(\alpha-\beta)-\cos(\alpha+\beta)\right)\implies$$
$$\sin\frac{2x}{3^n}\sin\frac x{3^n}=\frac12\left(\cos\frac x{3^n}-\cos\frac x{3^{n-1}}\right)\implies$$
$$\sum_{n=1}^\infty\sin\frac{2x}{3^n}\sin\frac x{3^n}=\frac12\left(\cos\frac x3-\cos x+\cos\frac x9-\cos\frac x3+\cos\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Matrix Power Formula
Prove that for a fixed $a \in \mathbb{R}$ we have the matrix power formula for all $n \in \mathbb{Z}_+$:
$$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^n = \begin{pmatrix}a^n & na^{n-1}\\0 & a^n\end{pmatrix}$$
How would we prove this? Right now we are doing work with proofs by induction... but how ... | Verify that it is true for $n = 1$. This is the induction basis.
Then assume the result holds for $n$, so
$$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^n = \begin{pmatrix}a^n & na^{n-1}\\ 0 & a^n\end{pmatrix}$$
This is the induction hypothesis.
Finally, show (with the help of the induction hypothesis) that the result hold... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/829633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity) Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity
$$
\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}
$$
holds?
| Generating function can do the job quite easily:
\begin{align*}
\frac{1}{(1-x)^k} &= \sum_{i\ge 0} \binom{i+k-1}{k-1}\, x^i
\end{align*}
Using convolution of generating functions,
\begin{align*}
\frac{1}{(1-x)}\cdot \frac{1}{(1-x)^k} &= \sum_{n\ge 0} \left(\sum_{i=0}^n \binom{i+k-1}{k-1}\right) x^n \\
\frac{1}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/833451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 3
} |
Locate my error for this initial value separable differential equation? The problem is to solve $ sin\,2x\,dx + cos\,3y\,dy = 0, \;\;\;\;y({\pi\over 2}) = {\pi\over 3}$
Here are my steps:
$$ cos \,3y \,dy = -sin \,2x \,dx $$
$$\int cos\,3y\,dy = \int -sin\,2x\,dx$$
$$ {1\over 3}\,sin\,3y = {1\over 2}\,cos\,2x\,+C $$
$$... | The equation
\begin{align}
- \sin(2x) \ dx = \cos(3y) \ dy
\end{align}
can be seen to have the solution
\begin{align}
\frac{1}{2} \cos(2x) = \frac{1}{3} \sin(3y) + c.
\end{align}
This can be seen in the form
\begin{align}
\sin(3y) = \frac{3}{2} \left( \cos(2x) - 2 c \right)
\end{align}
or
\begin{align}
y(x) = \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/833523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Help Evaluating $\lim_{x \to \infty} \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}$ Does anyone know how to evaluate the following limit?
$$
\lim_{x \to \infty} \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}
$$
The answer is $\frac{1}{2}$, but I want to see a step by step solution if possible.
| Using Taylor expansion:
$$\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}$$
$$\sqrt{x + \sqrt{x}+\dfrac{1}{2} + O(\frac{1}{\sqrt{x}})} - \sqrt{x}$$
$$\sqrt{x}+\dfrac{1}{2\sqrt{x}}(\sqrt{x}+\dfrac{1}{2}+ O(\frac{1}{\sqrt{x}})) + O(\frac{1}{\sqrt{x}})- \sqrt{x}$$
$$\dfrac{1}{2} + \dfrac{1}{4\sqrt{x}}$$
Which in the limit case ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/834471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Stuck on Indefinite Integral Please help me. I have been stuck on this for ages :(
$$\int \frac{1}{13\cos x+ 12}\,\mathrm{d}x$$
I appreciate any and all help. Thank you.
| $\bf{My\; Solution::}$ Given $$\displaystyle \int\frac{1}{13\cos x+12}dx = \int\frac{1}{13(1+\cos x)-1}dx$$
Now Using $$\displaystyle 1+\cos x = 2\cos^2 \frac{x}{2}\;,$$ we get
$$\displaystyle \int\frac{1}{26\cos^2 \frac{x}{2}-1}dx$$
Now Divide both $\bf{N_{r}}$ and $\bf{D_{r}}$ by $\displaystyle \cos^2 \frac{x}{2}$
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/834765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
What is the probability that the difference of squares of two positive integers up to $30$ is divisible by $3$ or $7$? If we choose any two numbers $a$ and $b$ from the integers $1$ through $30$, what is the probability of $a^2-b^2$ of being divisible by $3$ or $7$?
| Note that $a^2-b^2$ is divisible by $c$ if and only if
$$a\equiv b\equiv0\pmod{c}\qquad\text{ or }\qquad a\equiv\pm b\not\equiv0\pmod{c}.$$
Taking $c=3$ we find that the sum of the probabilities of these cases is
$$\left(\frac{10}{30}\right)^2+\left(\frac{20}{30}\right)^2=\frac{500}{900}.$$
For $c=7$ there are $9$ inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/838345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Help understanding this approximation In a paper that I'm reading, the authors write:-
$$N_e \approx \frac{3}{4} (e^{-y}+y)-1.04. \tag{4.31}$$
Now, an analytic approximation can be obtained by using the expansion
with respect to the inverse number of "e-foldings" ($N_e$ is the
number of "e-foldings"). For instance... | The assumption seems to be that $y \ll -1$ and $N_\epsilon \gg 1$. For brevity let's set $x = e^y$ and $N_\epsilon = N$ so that the equation becomes
$$
N = \frac{3}{4 x} + \frac{3}{4}\ln x - 1.04. \tag{1}
$$
The condition $y \ll -1$ now corresponds to $0 < x \ll 1$ for the new variable. In this case the term involvin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/839812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solutions of a system of linear equations with a parameter I've come across a linear algebra problem that I'm not sure how to solve. It's a generic problem - I have to find the solutions of a system of equations dependent on a parameter. So, my first though was to check for which values of the parameter the matrix on t... | $$\left(\begin{array}{ccc|c}
\lambda & 1 & 1 & 1 \\
1 & \lambda & 1 & \lambda \\
1 & 1 & \lambda & \lambda^2
\end{array}\right)\sim
\left(\begin{array}{ccc|c}
\lambda+2 & \lambda+2 & \lambda+2 & 1+\lambda+\lambda^2 \\
1 & \lambda & 1 & \lambda \\
1 & 1 & \lambda & \lambda^2
\end{array}\right)\overset{(1)}\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Proving that $\gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $. In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is
$$
\gamma
= \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)}
~ \mathrm{d... | We want use the transformation formula.
\begin{align*}
&U:= \left\lbrace (t,r) \mid t \in (0,1), r \in (t,1) \right\rbrace , V:= (0,1)^2,
\\ &f: V \rightarrow \mathbb{R}, \left( \begin{array}{r}
x\\
y\\
\end{array} \right) \mapsto \frac{x - 1}{(1-xy) \cdot ln(xy)},
\\ &\varphi: U \rightarrow V, \left( \begin{array}{r}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 4,
"answer_id": 3
} |
Is $x^8 + x^5 + x^3 + x^2 + 1$ an irreducible polynomial or not in GF $2^8$ Is $x^8 + x^5 + x^3 + x^2 + 1$ an irreducible polynomial or not in Galois Field $2^8$? Thanks in advance.
| The question is a bit unclear. If you are truly asking whether this polynomial
$$
p(x)=x^8+x^5+x^3+x^2+1
$$ is irreducible in the ring $GF(256)[x]$, then the answer is trivially NO! This is because:
*
*Either it is reducible over $GF(2)$ in which case it is reducible over the extension field also,
*Or it is irreduc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to solve the recurrence $T(n) = T(n-1) +\sqrt{n}$? While solving the recurrence of the title I come to the series
$$T(n) = \sqrt1 + \sqrt2 + \sqrt{3} + \cdots + \sqrt n.$$
Please somebody help me how to solve this.
| We can use left-hand and right-hand approximations of integrals via Riemann sums to obtain tight bounds on our desired sum. Since $f(x) = \sqrt{x}$ is a strictly increasing function, observe that:
\begin{align*}
\int_0^n \sqrt{x} \, dx \leq \sum_{k=1}^n \sqrt{k} &\leq \int_1^{n+1} \sqrt{x} \, dx \\
\left[ \frac{x^{3/2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/845490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Range of the function $f(x) = \frac{x^2+14x+9}{x^2+2x+3}\;,$ where $x\in \mathbb{R}$ Calculation of Range of the function $\displaystyle f(x) = \frac{x^2+14x+9}{x^2+2x+3}\;,$ where $x\in \mathbb{R}$
(Can we solve it Using $\bf{A.M\geq G.M}$) Inequality.
$\bf{My\; Try::}$ Let $\displaystyle y = f(x) = \frac{x^2+14x+9}{x... | HINT:
Let $\displaystyle y=\frac{x^2+14x+9}{x^2+2x+3}$
Rearrange to form a Quadratic Equation in $x$
As $x$ is real, the discriminant must be $\ge0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/845952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Working with groups. Finding the inverse of some $S_9$ I want to compute the inverse of:
$\begin{pmatrix} 1&2&3&4&5&6&7&8&9\\3&2&1&6&5&9&4&8&7 \end{pmatrix}$
Sorry about alignment(they are all one digit though)
Now it would seem that the inverse here is itself. But I feel like maybe I am just confusing myself now, and ... | Each element in $S_9$ is a permutation (or bijection) $\{1, 2, \dots, 9\} \to \{1, 2, \dots, 9\}$. In particular, your permutation I can denote as a function $f$, where:
$$\begin{align}
f(1) &= 3\\
f(2) &= 2 \\
f(3) &= 1 \\
f(4) &= 6 \\
f(5) &= 5 \\
f(6) &= 9 \\
f(7) &= 4 \\
f(8) &= 8 \\
f(9) &= 7\text{.}
\end{align}$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/846598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Finding closed form for $1^3+3^3+5^3+...+n^3$ I'd like to find a closed form for $1^3+3^3+5^3+...+n^3$ where $n$ is an odd number.
How would I go about doing this?
I am aware that $1^3+2^3+3^3+4^3+...+n^3=\frac{n^2(n+1)^2}{4}$ but I'm not too sure how to proceed from here.
My gut feeling is telling me to multiply the ... | Using repeated differences we get
$$
\begin{array}{llll}
1 & 28 & 153 & 496 & 1225 & 2556 & \\
27 & 125 & 343 & 729 & 1331 & \\
98 & 218 & 386 & 602 & \\
120 & 168 & 216 & \\
48 & 48 & \\
0 & \\
\end{array}
$$
Newton's interpolation formula then gives us
$$
1 \binom{n-1}{0} + 27 \binom{n-1}{1} + 98 \binom{n-1}{2} + 120... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/848087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 1
} |
Evaluation of $ \int\frac{\sin (x+\alpha)}{\cos^3 x}\cdot \sqrt{\frac{\csc x+\sec x}{\csc x-\sec x}}dx$ Evaluation of $\displaystyle \int\frac{\sin (x+\alpha)}{\cos^3 x}\cdot \sqrt{\frac{\csc x+\sec x}{\csc x-\sec x}}dx$
$\bf{My \; Try::}$ Let $\displaystyle I = \int \frac{\sin (x+\alpha)}{\cos^3 x}\cdot \sqrt{\frac{\c... | Assuming what you have done up till the last part is correct, these hints I think will help you go further:
(1) Multiplying and dividing by $\sqrt{1+t}$
(2) Knowing the derivative of $\sinh^{-1} (\sqrt{t-1}) $ and $\sqrt{t+1}\sqrt{t-1}$
(3) Using partial integration
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/853888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$ I am currently attempting to prove the following inequality
$\dfrac{a}{b+c}+\dfrac{b}{a+c}+\dfrac{c}{a+b} \geq \dfrac{3}{2}$ for all $ a,b,c>0$
My instinctive plan of attack is to use the AM/GM inequality with $x_1=\dfrac{a}{b+c}$ et... | Here is another nice proof.
We have to prove;
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq \frac{3}{2}\\\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}\geq \frac{3}{2}+3\\(a+b+c)\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)\geq \frac{9}{2}\\([b+c]+[a+c]+[a+b])\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/855283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 3
} |
Proof of $\sin2x+x\sin^2x \lt\dfrac{1}{4}x^2+2$ How can be proven the following inequality?
$$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
Thanks
| $$\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
$$\sin(x)(2\cos(x)+x\sin(x))\lt\dfrac{1}{4}x^2+2$$
while the right side is positive for all $x \in \mathbb{R}$ it is enoght that we show:
$$|\sin(x)|\cdot|(2\cos(x)+x\sin(x))|\lt\dfrac{1}{4}x^2+2$$
by the Cauchy–Schwarz inequality we have:
$$|\sin(x)|\cdot\sqrt{|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/856237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to solve: $\frac{2^{n+1}}{n+1}=\frac{4+2^n}{3}$ $n$ is an integer variable satisfying $$\frac{2^{n+1}}{n+1}=\frac{4+2^n}{3}$$ How can I find $n$?
| Rewriting, we have $3\cdot2^{n+1}=(n+1)(4+2^n)$
First, note that whenever $n\geq 5$, we have $(n+1)(4+2^n)>6\cdot 2^n=3\cdot2^{n+1}$
Next note that whenever $n<-1$, $(n+1)(4+2^n)<0<3\cdot2^{n+1}$
Note that $n=-1$ isn't possible because in the first equation we would divide by $0$.
Hence any solution must have $n\in[0,4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/859805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$ Question: Prove that if
$$a_n=\left\{
\begin{array}{ll}
a_1=1\\
a_{n+1}=1+\frac{1}{1+a_n}
\end{array}
\right.$$
then $a_n$ converges, and then find $\lim_{n \to \infty}a_n$.
I found that $\lim_{n \to \infty}a_n=\sqrt{2}$, and that $1\leq a_n<2$, but the... | Given
$$
a_{n+1} = 1 + \frac{ 1 }{ 1 + a_n }.
$$
Which can be written as
$$
a_{n+1} = \frac{ 2 + a_n }{ 1 + a_n }.
$$
Write
$$
a_n = \frac{ P_n }{ Q_n },
$$
then
$$
a_{n+1} = \frac{ 2 + a_n }{ 1 + a_n } =
\frac{ \displaystyle 2 + \frac{ P_n }{ Q_n } }{ \displaystyle 1 + \frac{ P_n }{ Q_n } }
= \frac{ P_n + 2Q_n }{ P_n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 9,
"answer_id": 2
} |
Fair Coin Flips
Michele flips a fair coin nonstop. Two students, Thomas and George,
decide to
make a bet about whose sequence of flips will occur first from the moment they
begin observing the results of Michele's flips. Thomas picks the sequence 'HTT'.
Find, with proof, a sequence George can pick that gives... | First, let us see the expected number of tosses required for each sequence of three:
\begin{align*}
\begin{array}{|c|c|}\hline
\mathrm{HHH}, \mathrm{TTT} & 14\\
\mathrm{HHT}, \mathrm{THH}, \mathrm{TTH}, \mathrm{HTT} & 8\\
\mathrm{HTH}, \mathrm{THT} & 10 \\ \hline
\end{array}
\end{align*}
Since $\mathr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)
Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer.
I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
| That $\,47\,$ is a factor can be verified by very easy mental arithmetic:
$\begin{eqnarray}
{\rm mod}\ 47\!:\quad && 1+ \color{#c00}{7^8} +\, 8^9 +\ \color{#0a0}{9^7}\\
\equiv && 1+ \color{#c00}{2^4} + 2^{27} + \color{#0a0}{7\cdot 7^6\cdot 8^7}\quad {\rm by}\ \ \color{#0a0}{9\equiv 7\cdot 8},\ \ \color{#c00}{7^2\equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Definite trigonometric integral This question is motivated by Iterative Mean, Covariance Algorithm Convergence:
Is there a closed form for the integral
$$
\int_0^{2 \pi} \frac{\sin^n(\theta)\cos^m(\theta)}{A\sin^2(\theta)+B\cos^2(\theta)+2C\sin(\theta)\cos(\theta)} d\theta\,?
$$
where $AB \geq C^2$ and $m,n \in \{0,1,... | Dividing the numerator and denominator by $\cos^2 \theta$ we get
$$I = \int_0^{2\pi } {\frac{{{{\sin }^n}\theta {{\cos }^{m - 2}}\theta }}{{A{{\tan }^2}\theta + 2C\tan \theta + B}}d\theta }.$$
Now let us solve the equation $$A{{\tan }^2}\theta + 2C\tan \theta + B=0$$ for simplicity we let $u=\tan \theta$, so that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find odd numbers $(o_1,o_2,o_3,o_4)$ such that $o_1^2-o_2^2=2(o_3^2-o_4^2)$ such that $o_1>o_2$ and $o_3>o_4$ I am working on a graph labeling problem and am stuck at the following problem on odd numbers.
Find (all) odd numbers $(o_1,o_2,o_3,o_4)$ such that $o_1^2-o_2^2=2(o_3^2-o_4^2)$ such that $o_1>o_2$ and $o_3>o_4$... | Given the condition, let use write
\begin{eqnarray}
o_1 &=& o,\\
o_2 &=& o - 4p,\\
o_3 &=& o - 2q,\\
o_4 &=& o - 2 r.
\end{eqnarray}
Then we get
$$
o\Big(p+q-r\Big)=2p^2+q^2-r^2.
$$
Case 1
When
$$
p+q-r = 0,
$$
we obtain
$$
p \Big( p -2q \Big) = 0,
$$
so we obtain
\begin{eqnarray}
o_1 &=& 1 + 2 v + 8 w,\\
o_2 &=& 1 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
What is the value of $ \int_{x}^{1} \arcsin \left( \frac{2t}{t^2+1} \right) \text{d}t $? Is this result true? Wolfram doesn't seem to be able to evaluate the definite integral in the allowed time. $$ \int_{x}^{1} \arcsin \left( \dfrac{2t}{t^2+1} \right) \text{d}t = \dfrac{\pi}{2} - 2x\arctan x - \log(2) + \log(1+x^2) $... | $$
\frac{2t}{1+t^2} = \frac{2\tan\theta}{1+\tan^2\theta} = \frac{2\tan\theta}{\sec^2\theta} = 2\sin\theta\cos\theta = \sin (2\theta).
$$
So
$$
\arcsin\left(\frac{2t}{1+t^2}\right) = 2\theta.
$$
$$
dt = \sec^2\theta\,d\theta
$$
As $t$ goes from $x$ to $1$, then $\theta=\arctan t$ goes from $\arctan x$ to $\pi/4$.
$$
\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/875227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Find the equation of the linear transformation of an orthogonal projection on the line y=mx. Let $T : \mathbb R^2 → \mathbb R^2$ the orthogonal projection on the line $y = mx$. Prove that for all $a, b \in \mathbb R$,
$$\begin{align}T((a,b)) = {\frac{1}{m^2 + 1}}(a+mb, ma + m^2b)\end{align}$$
The mark scheme talks abou... | Here's how I solve this problem:
Notice I am writing vectors in columnar form; thus, the OP's $(a, b)$ is my
$\begin{pmatrix} a \\ b \end{pmatrix} \tag{0}$
etc.; these things being said , we have:
The line $y = mx$ is in fact the set of points (vectors) $L = \{(x, mx)^T = x(1, m)^T \mid x \in \Bbb R \}$; this is the on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve $\sin^2 x − \cos^2 x = \sin x$, when $x\in -\pi ≤ x ≤\pi$ I have to solve for $x$ using the domain of $-\pi ≤ x ≤\pi$
$$\sin^2 x − \cos^2 x = \sin x $$
I tried changing $\cos^2 x$ to $1 - \sin^2 x$, and then getting
$$\sin^2 x - 1 + \sin^2 x = \sin x \to 2\sin^2 x - 1 = \sin x$$
Then I have no clue where to go ... | From
$\sin^2 x − \cos^2 x = \sin x$,
since
$\cos^2 x = 1-\sin^2 x$,
$\sin x
= \sin^2 x - (1-\sin^2 x)
=2\sin^2 x -1
$.
Now we only have $\sin x$,
so let $\sin x = y$.
This becomes
$y = 2y^2-1$.
You can solve this for $y$.
From the possible values of $y$,
you can then get $x$.
I will leave it at that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/877506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there also an other way to show the equality: $\left\lfloor \frac{n}{2}\right\rfloor + \left\lceil \frac{n}{2} \right\rceil=n$? I want to show that:
$$ \left\lfloor \frac{n}{2}\right\rfloor + \left\lceil \frac{n}{2} \right\rceil=n$$
That's what I have tried:
*
*$ \left\lfloor \frac{n}{2}\right\rfloor=\max \{ m \i... | For even $n$, $\left\lfloor\frac n2\right\rfloor=\left\lceil\frac n2\right\rceil=\frac n2$.
For odd $n$, $\left\lfloor\frac n2\right\rfloor=\frac{n-1}2$, and $\left\lceil\frac n2\right\rceil=\frac{n+1}2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/878854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Matrix exponential of a skew-symmetric matrix without series expansion I have the following skew-symmetric matrix
$$C = \begin{bmatrix} 0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0 \end{bmatrix}$$
How do I compute $e^{C}$ without resorting to the series expansion... | Let $x = \sqrt{a_1^2+a_2^2+a_3^2}$. You can verify that $C^3 = -(a_1^2+a_2^2+a_3^2)C = -x^2C$.
Hence, $C^{2m+1} = (-1)^mx^{2m}C$ and $C^{2m} = (-1)^{m-1}x^{2m-2}C^2$.
Therefore, $e^C = \displaystyle\sum_{n=0}^{\infty}\dfrac{1}{n!}C^n = I + \sum_{m=0}^{\infty}\dfrac{1}{(2m+1)!}C^{2m+1} + \sum_{m=1}^{\infty}\dfrac{1}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 1
} |
Indefinite integral of trignometric function What is the trick to integrate the following
$$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
| Using $$\cos2A=\frac{1-\tan^2A}{1+\tan^2A}$$
$$\int\frac{1-\cos x}{\cos x(1+\cos x)}dx$$
$$=\int\frac{2\sin^2\dfrac x2}{\cos^2\dfrac x2-\sin^2\dfrac x2}\frac12\sec^2\dfrac x2dx$$
$$=\int\frac{2\tan^2\dfrac x2}{1-\tan^2\dfrac x2}\frac12\sec^2\dfrac x2dx$$
$$=\int\frac{2u^2-2+2}{1-u^2}du=-2u+2\int\frac{du}{1-u^2}du$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/880315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Trignometry-Prove that $(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$ Prove that $$(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$$
I tried solving the LHS and RHS seperately but they were not coming ... | $$\begin{array}{lll}
(\csc\theta - \sec\theta)(\cot\theta - \tan\theta)&=&(\csc\theta - \sec\theta)\bigg(\frac{\csc\theta}{\sec\theta} - \frac{\sec\theta}{\csc\theta}\bigg)\\
&=&(\csc\theta - \sec\theta)\bigg(\frac{\csc^2\theta-\sec^2\theta}{\sec\theta\csc\theta}\bigg)\\
&=&\frac{(\csc\theta-\sec\theta)(\csc\theta-\sec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
prove $\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$
If $x$,$y$,$z$ are positive real numbers,Prove:$$\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$$
Using this two inequality:
$\sum ^n_{i=1} \sqrt{a_ib_i}\leq\sqrt {ab} $ (we call it $A$ inequality)
$\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_... | By your work and by C-S we obtain:
$$\sum_{cyc}\frac{x}{x+\sqrt{(x+y)(x+z)}}\leq\sum_{cyc}\frac{x}{2x+\sqrt{yz}}=\frac{3}{2}-\sum_{cyc}\left(\frac{1}{2}-\frac{x}{2x+\sqrt{yz}}\right)=$$
$$=\frac{3}{2}-\frac{1}{2}\sum_{cyc}\frac{\sqrt{yz}}{2x+\sqrt{yz}}=\frac{3}{2}-\frac{1}{2}\sum_{cyc}\frac{yz}{2x\sqrt{yz}+yz}\leq$$
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/882654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find an equation for a moving rod The two endpoints of a 1-metre long rod have an initial position at $(0,0),(0,1).$ The rod slides continuously to the position $(1,0),(0,0)$ sweeping out a region in the positive quadrant. Determine the equation for the boundary of this region.
My attempt:
At any position, let $a$ den... | Firstly obtain the straight-line equation of the rod at any point in the slide. If $x_0$ and $y_0$ are the $x$-intercept and $y$-intercept, then by Pythagoras, $x_0^2 + y_0^2 = 1$.
So we have the equation of this line:
\begin{eqnarray*}
y &=& -\dfrac{y_0}{x_0}x + y_0 \\
&=& -\dfrac{y_0}{\sqrt{1 - y_0^2}}x + y_0
\end{eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/884742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find X location using 3 known (X,Y) location using trilateration I post this question in stackoverflow here and was advised it was best suited for here.
I am trying to understand the maths behind trilateration, we have 3 access points $(\text{AP's: } 1,2,3)$ and we know the centre coordinates of theses $3$ (AP's). Lost... | Use the distance equation. If your unknown point is $(x,y)$, your known points are $(x_i, y_i)$ which are distances $r_i$ from your unknown point, then you get three equations:
$$(x-x_1)^2 + (y-y_1)^2 = r_{1}^2 \\ (x-x_2)^2 + (y-y_2)^2 = r_{2}^2 \\ (x-x_3)^2 + (y-y_3)^2 = r_{3}^2$$
We can expand out the squares in eac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/884807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
How do I go from this $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$? So I am doing $\int\frac{x^2-3}{x^2+1}dx$ and on wolfram alpha it says the first step is to do "long division" and goes from $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$. That made the integral much easier, so how would I go about doing that in a clear... | Since the residue in $x=i$ of $\frac{x^2-3}{x^2+1}$ is given by:
$$\lim_{x\to i}\frac{x-i}{x^2+1}(x^2-3)=(i^2-3)\lim_{x\to i}\frac{1}{2x}=2i$$
due to De l'Hospital theorem, and the residue in $x=-i$ is just the opposite due to the Schwarz reflection principle, we have that:
$$f(x)=\frac{x^2-3}{x^2+1}-\left(\frac{2i}{x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/885999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 6
} |
Prove that $ \frac {12^{x-2}.4^{x}} {6^{x-2}} = 2^{3x-2} $ Can someone please help me with this question?
$ \large \frac {12^{x-2}.4^{x}} {6^{x-2}} = 2^{3x-2} $
My steps so far:
$ \large \frac {4^{x-2}.3^{x-2}.4^{x}}{3^{x-2}.2^{x-2}} = 2^{3x-2} $
$ \large \frac {4^{x-2}.4^{x}}{2^{x-2}} = 2^{3x-2} $
$ \large \frac {4^{2... | $$\large \frac {12^{x-2}.4^{x}} {6^{x-2}}$$
$$\begin{align}
& \implies \large \frac {(3\times4)^{x-2}.2^{2x}} {(3\times2)^{x-2}} \\
& \implies \large \frac {(3)^{x-2}\times(2)^{2(x-2)}\times2^{2x}} {(3\times2)^{x-2}} \\
& \implies \large \frac {(3)^{x-2}\times(2)^{2(x-2)}2^{2x}} {(3)^{x-2}\times(2)^{x-2}} \\
& \implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Evaluation of limit How to evaluate the value of this limit?
$$\lim_{x\to 2} \frac{\sqrt{x-2} + \sqrt x - \sqrt2}{\sqrt{x^2 - 4}}$$
Actually I'm struck at algebraic part. Please guide..
| Deepak's answer is better, but you can also do as the followings :
$$\begin{align}\frac{(\sqrt{x-2}+\sqrt x)-\sqrt 2}{\sqrt{x^2-4}}\cdot \frac{(\sqrt{x-2}+\sqrt x)+\sqrt 2}{(\sqrt{x-2}+\sqrt x)+\sqrt 2}\end{align}$$
$$=\frac{x-2+2\sqrt{x(x-2)}+x-2}{\sqrt{(x-2)(x+2)}\ (\sqrt{x-2}+\sqrt x+\sqrt 2)}$$
$$=\frac{\color{red}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
If $\gcd(a,b)=1$, then every odd factor of $a^2 + 3b^2$ has this same form EDIT: Please see EDIT(2) below, thanks very much.
I want to prove by infinite descent that the positive divisors of integers of the form $a^2+3b^2$ have the same form. For example, $1^2+3\cdot 4^2=49=7^2$, and indeed $7,49$ can both be written i... | Note that:
$$(ax-3by)^2 + 3(ay+bx)^2 = (a^2+3b^2)(x^2+3y^2) = (ax+3by)^2 + 3(ay-bx)^2$$
So if all divisors of numbers of that form are also of that form, then only numbers of that form that are primes have unique representation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/888272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Solve $(1+z)^8=(1-z)^8$ My guess is to write this as $$\left(\frac{1+z}{1-z}\right)^8=1.$$ We can then find 8 possibilities for $\frac{1+z}{1-z}$, namely $\cos(k\pi/4)+i\sin(k\pi/4)$, $k=1,\ldots,8$. For each $k$ we can then deduce 2 equations by putting $z=x+iy$, for example for $k=1$ we get: $$\frac{1+x+iy}{1-x-iy}=\... | HINT:
$\left(\dfrac{1+z}{1-z}\right)^8=1 \implies$
$\left(\dfrac{1+z}{1-z}\right)^4=\pm{1} \implies$
$\left(\dfrac{1+z}{1-z}\right)^2=\pm{1},\pm{i} \implies$
$\left(\dfrac{1+z}{1-z}\right)^1=\pm{1},\pm{i},\pm{\dfrac{1+i}{\sqrt{2}}},\pm{\dfrac{1-i}{\sqrt{2}}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/890009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Interesting combinatorial identities Let $n$ be a strictly positive integer and let $j=0,\dots,n-1$.
By using Mathematica I managed to guess the following identities:
\begin{eqnarray}
\sum\limits_{m=0}^{n-j-1} \binom{n-m-1}{j} \binom{n+m}{j} &=& \frac{1}{2} \binom{2 n}{2j + 1} \\
\sum\limits_{m=0}^{n-j-1} \binom{n-m-1}... | Hint: The first identity can be explained as follows.
One wants to count how many sequences $a_1<\cdots<a_{2j+1}$ can be formed from $\{1,\dots,2n\}$. Of course the standard answer is ${2n}\choose{2j+1}$.
Let's count differently by looking at the middle number $a_{j+1}$. It can only take one of the values $j+1,\dots, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Globally Lipschitz condition
Show that the function $f(x)=\frac1{1+x^2}$ satisfies a global
Lipschitz condition on $\mathbb R$.
By the definition we want to whow that $|f(x)-f(y)|\leqslant L|x-y|$. Here is my work so far.
$$\left|\frac1{1+x^2}-\frac1{1+y^2}\right|=\left|\frac{y^2-x^2}{(1+y^2)(1+x^2)}\right|\leqslan... | $$\left| \frac{x+y}{(1+x^2)(1+y^2)}\right|=\frac{|x+y|}{(1+x^2)(1+y^2)}$$
By triangle inequality:
$$\frac{|x+y|}{(1+x^2)(1+y^2)} \leq \frac{|x|}{(1+x^2)(1+y^2)}+\frac{|y|}{(1+x^2)(1+y^2)}$$
Now $x^2+1 \geq 1$, so:
$$ \frac{|x|}{(1+x^2)(1+y^2)} \leq \frac{|x|}{1+x^2}$$
Next $\displaystyle \frac{|x|}{1+x^2} \leq 2$ be AM... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the value of $27\csc^2\theta+8\sec^2\theta$ $10\sin^4\theta+15\cos^4\theta=6$, then find the value of $27\csc^2\theta+8\sec^2\theta$
I don't know how to do it have just tried by converting sin and cos into csc and sec. But can't get the answer.
| Since $\sin^2\theta+\cos^2\theta=1$,
$$10\sin^4\theta+15\cos^4\theta=6(\sin^2\theta+\cos^2\theta)^2.$$
It follows that
$$4\sin^4\theta-12\sin^2\theta\cos^2\theta+9\cos^4\theta=0,$$
or
$$(2\sin^2\theta-3\cos^2\theta)^2=0.$$
Thus we have $2\sin^2\theta=3\cos^2\theta$. Can you continue from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/890738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.