Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Proof polynomial has only one real root. I need to prove that this polynomial equation:
$$x^5-(3-a)x^4+(3-2a)x^3-ax^2+2ax-a=0\quad\text{ for }\quad a\in(0,\frac{1}{2}).$$
has only one root. That it has one real root is obvious because it is of odd degree. But Descartes rules here fails to bound the number of roots to o... | Consider
$$p(x) = x^5-(3-a)x^4+(3-2a)x^3-ax^2+2ax-a$$
First, we note that if $x< 0$, each term is negative, hence there are no negative roots. Also, $p(0) = -a < 0$. Further,
$$p'(x) = 5x^4-4(3-a)x^3+3(3-2a)x^2-2ax+2a$$
So it is sufficient to show that $p'(x) > 0$ for $x > 0, \; a \in (0, \frac12)$.
For this, note... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$ This is supposed to be an application of AM-GM inequality.
if $abc=1$, then the following holds true: $a^2+b^2+c^2\ge a+b+c$
First of all,
$a^2+b^2+c^2\ge 3$
by a direct application of AM-GM.Also,we have
$a^2+b^2+c^2\ge ab+bc+ca$
Next,we consider the expression
$(a+1)(b+1)(c+1)... | Here is an exotic solution based on geometry.
Let $\mathcal{M}$ and $\mathcal{S}$ be surfaces defined by
\begin{align*}
\mathcal{M} : abc = 1
\quad \text{and} \quad
\mathcal{S} : a^{2} + b^{2} + c^{2} = a + b + c.
\end{align*}
Then we have the following observations:
*
*$\mathcal{M}$ lies outside the sphere of radiu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How was this solution found? Consider an empty spherical bowl of radius $r$. I was trying to find the height to which I would need to fill the bowl with water so that it would be one quarter full (in terms of volume).
The total volume is $\frac{4}{3}\pi r^3$ and the volume filled be water up to a height $h$ is $\pi r h... | After some reading around, I think that I have found the solution.
Starting with $h^3-3rh^2+r^3=0$, we make the substitution $h=x+r$ to get the reduced cubic $x^3 -3r^2x-r^3$, or equivalently $x^3 = 3r^2x+r^3$.
Next, we make the substitution $x=2r\cos\theta$ which yields $8r^3\cos^3\theta = 6r^3\cos\theta + r^3$. Sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Use the integration formula $\frac{1}{a}\arctan\frac{x}{a}$ to solve $\frac{1}{2} \int_{-1}^1 \mathrm{ \frac{dx}{1+\sqrt{2}x+x^2} }\, $ As question states, I am trying to figure out how to use the integration formula to solve the integral. My issue is that the integral isn't of the form $\frac{dx}{a^2+x^2}$
| Hint: $$x^2\pm\sqrt{2}x+1 = (x\pm 1/\sqrt{2})^2+1/2$$
details:
$$\int_{-1}^1 \frac {dx}{x^2+\sqrt{2}x+1}
= \int_{-1}^1 \frac {dx}{(x+ 1/\sqrt{2})^2+1/2}
= \int_{-1+1/\sqrt{2}}^{1+1/\sqrt{2}}
\frac {du}{u^2+1/2}
\\
= \left[1/\sqrt{2} \arctan \sqrt{2} u
\right]_{-1+1/\sqrt{2}}^{1+1/\sqrt{2}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/741497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How to find the indefinite integral $\int \frac{dx}{1+x^{n}}$? How to find the indefinite integral $$\int \frac{dx}{1+x^{n}}$$ where n is a positive integer?
| I just wanted to comment to give the general formula for this explicitly in case you have interest, but I can't comment so I must post an answer. For $n=2q-1,$ with $q\in\mathbb{N},$ and $m<n$ both natural numbers,
$$\int\frac{x^{m-1}}{x^n+1}dx=\frac{\left(-1\right)^{m-1}}{n}\log\left(x+1\right)\\-\frac{1}{n}\sum\limit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Calculating $1+\frac13+\frac{1\cdot3}{3\cdot6}+\frac{1\cdot3\cdot5}{3\cdot6\cdot9}+\frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $
How to find infinite sum How to find infinite sum $$1+\dfrac13+\dfrac{1\cdot3}{3\cdot6}+\dfrac{1\cdot3\cdot5}{3\cdot6\cdot9}+\dfrac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\do... | \begin{align*}
1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+
\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+\ldots
&=\sum_{n=0}^{\infty}
\frac{(2n-1)!!}{3^{n} n!} \\
&=\sum_{n=0}^{\infty}
\frac{(-\frac{1}{2})(-\frac{3}{2})\ldots (-\frac{2n-1}{2})}
{3^{n} n!} (-2)^{n} \\
&=\sum_{n=0}^{\infty} \binom{-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/746388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
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"answer_id": 0
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p odd prime. Prove that if $a\equiv b\pmod p$ then $a^p\equiv b^p\pmod p^2$. Then show $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$ Question:
Let $p$ be an odd prime. Prove that if $a\equiv b \pmod p$ then $a^p\equiv b^p \pmod p^2$. Then show the Diophantine equation $x^5+y^5=z^5$ has no integer solution... | Write the Fermat equation of exponent $5$ in the form $x^5+y^5+z^5=0$.
We have $x^5\equiv x \mod 5$, $y^5\equiv y \mod 5$ and $z^5\equiv z \mod 5$, hence $x+y+z\equiv 0\mod 5$. Without loss of generality we may assume, because $5\nmid xyz$, that $x\equiv y\mod 5$, hence $x^5\equiv y^5 \mod 25$, and $−z^5 ≡ x^5 + y^5 ≡ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/746590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Help with telescoping sum $\sum_{i=3}^n \frac{1}{i(i+3)} $ How can I use the telescoping technique to compute the following sum?
I'm having issues getting started. I know the basic steps but I don't know how to perform them. I know I have to separate the fraction into A and B. After that I have to perform the sum but I... | $\displaystyle\sum_{i=3}^n \dfrac{1}{i(i+3)}=\dfrac{1}{3}\sum_{i=3}^n \left(\dfrac{1}{i}-\dfrac{1}{i+3}\right)=\dfrac{1}{3}\sum_{i=3}^n \left(\dfrac{1}{i}-\dfrac{1}{i+1}+\dfrac{1}{i+1}-\dfrac{1}{i+2}+\dfrac{1}{i+2}-\dfrac{1}{i+3}\right)$
$\displaystyle=\dfrac{1}{3}\sum_{i=3}^n \left(\dfrac{1}{i}-\dfrac{1}{i+1}\right)+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/747822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove this inequation? $$
1+\frac{2}{3n-2}\leqslant \sqrt[n]{3}\leqslant 1+\frac{2}{n}, n\in \mathbb{Z}^{+}
$$
How to prove this inequation?
| For the LHS, apply $\text{AM} \ge \text{GM}$ to $n$ numbers with $n-1$ copies of $1$ and one copy of $\frac13$, we get
$$\begin{align}
&\left(1 + \frac{2}{3n-2}\right)^{-1}= 1 - \frac{2}{3n} = \frac{1}{n}\left( (n-1)\times 1 + \frac{1}{3}\right) \ge \frac{1}{\sqrt[n]{3}}\\
\implies &
1 + \frac{2}{3n-2} \le \sqrt[n]{3}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A little more integration. Help integrating
$$\int\cos^{-1}(a\tan\theta)\ d\theta$$
I understand that Wolfram gives a solution, but I'd like to know the steps. I haven't been able to rewrite the equation into anything helpful.
| Mathematica outputs $$\theta \cos ^{-1}(a \tan (\theta))+\frac{1}{4} \left(4 \theta \sin ^{-1}(a \tan (\theta))+i \left(2 i \left(\text{Li}_2\left(\left(\sqrt{a^2+1}-a\right) e^{-i \sin ^{-1}(a \tan (\theta))}\right)+\text{Li}_2\left(-\left(a+\sqrt{a^2+1}\right) e^{-i \sin ^{-1}(a \tan (\theta))}\right)\right)-2 i \lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/750158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove the inequality.Let a, b and c be nonnegative real numbers. Let $a$, $b$ and $c$ be nonnegative real numbers. Prove that $a^4+b^4+c^2\ge 8^{½}abc$
| Using AM,GM Inequality
$$a^4+b^4\ge 2a^2b^2$$ which can be demonstrated as $$a^4+b^4=(a^2-b^2)^2+2a^2b^2\ge 2a^2b^2$$ as the square of any real number is $\displaystyle\not<0$
Again similarly, $$2a^2b^2+c^2\ge 2\sqrt{2a^2b^2\cdot c^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/753719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $T(1)$, $T(x)$ and $T(x^{2})$ and $T(ax^2+bx+c)$ Let $$T:P_{3}\rightarrow P_{3}$$ be a linear transformation such that
$$T(2x^{2})=2x^{2}+3x, T(\frac{1}{2}x+2)=2x^{2}+4x-3, T(2x^{2}-1)=3x-1.$$
Find $$T(1)$$, $$T(x)$$ and $$T(x^{2})$$
and $$T(ax^{2}+bx+c)$$.
I will be completely honest and say that I have no idea ... | Here's one way:
given that
$T(2x^{2})=2x^{2}+3x, T(\frac{1}{2}x+2)=2x^{2}+4x-3, T(2x^{2}-1)=3x-1, \tag{0}$
start with $T(2x^2)$. We have, by linearity,
$T(2x^2) = 2T(x^2), \tag{1}$
and since we are given
$T(2x^2) = 2x^2 + 3x, \tag{2}$
we see by (1) that
$2T(x^2) = 2x^2 + 3x, \tag{3}$
or
$T(x^2) = x^2 + \dfrac{3}{2}x; ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Lagrange multipliers from hell I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way:
"Find the closest and furthest points on the circle made from the intersection of the ball $(x-1)^2+(y-2)^2+(z-3)^2=9$ and the plane $x-2z=0$ from the point $(0,0)$".
What I di... | The circle can be parametrized as
$$ x = 2 + \frac{4}{\sqrt 5} \sin \theta, \; y = 2 + 2 \cos \theta, \; z = 1 + \frac{2}{\sqrt 5} \sin \theta. $$
The squared distance of such a point from the origin is
$$ f(\theta) = 13 + 4 \sqrt 5 \sin \theta + 8 \cos \theta. $$
Derivative is
$$ f'(\theta ) = 4 \sqrt 5 \cos \the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/755510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Generating Functions for collection of balls There are 10000 identical red balls, 10000 identical yellow balls
and 10000 identical green balls. In how many different ways can we
select 2005 balls so that the number of red balls is even or the
number of yellow balls is odd?
in this question, being even or odd affect... | You are asking for generating functions.
First of all, the total number of balls available is irrelevant, as long as it is enough to fill all slots. For simplicity, take infinite number of balls.
*
*Red balls even: $1 + z^2 + \ldots = \frac{1}{1 - z^2}$
*Yellow balls odd: $z + z^3 + \ldots = \frac{z}{1 - z^2}$
*An... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate $\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$ I am trying to calculate:
$$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$
I am not looking for an answer but simply a nudge in the right direction. A strategy, just something that would get me started.
So, after doing the Taylor Expansion on the $\ln(1-x+x^2)$ ig to the follo... | Just to simplify the things, make the change of variables $s=2x-1$. The integral then reduces to
$$I=2\int_{-1}^1\frac{\ln\frac{3+s^2}{4}}{1-s^2}ds.\tag{1}$$
The antiderivative of any expression of the type $\displaystyle\frac{\ln P(x)}{Q(x)}$ is computable in terms of dilogarithms, essentially due to
$$\displaystyle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
| For any square matrix with one value on the diagonal and another value everywhere else, a consistent pattern of (orthogonal but not orthonormal) eigenvectors for the $n$ by $n$ case can be read from the columns of
$$
\left( \begin{array}{rrrrrrrrrr}
1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/757320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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The product of three consecutive natural numbers is divisible by $6$ Please give me feedback for my answer to this question.
Prove or find a counterexample: The product of any three consecutive
natural numbers is divisible by $6$
My answer: True. Suppose $n$ is a natural number, such that the $3$ consecutive natural... | You have proved that $1\times2\times3$ is divisible by six, not that the product of any 3 consecutive natural numbers is divisible by $6$.
If a number is divisible by $6$, then it must be divisible by both $2$ and $3$. Your product is $$n(n+1)(n+2)$$ so you could try showing that at least one of $n$, $n+1$ or $n+2$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/757393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Evaluate $\int x \sqrt{1 - x^4} \,\mathrm{d}x$ I have the following question
$$\int x \sqrt{1 - x^4} \,\mathrm{d}x$$
I know we have to use trig. substitution for this and therefore, I did the following by letting $x = \sin \theta$ and $dx = \cos \theta \,\mathrm{d}\theta$
\begin{align}
&\int x \sqrt{1-x^4} \,\mathrm{d}... | Try another substitution: $x^2 = \sin (u)$.
We have $2x dx = \cos (u) du$ so $dx = \frac{\cos (u)}{2x} du$
So now we have $\frac{1}{2} \displaystyle \int \cos(u) \sqrt{1 - \sin^2 (u)} du$
And I think you can do the rest.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/757809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$ Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$
$$
f(x) =
\left\{
\begin{array}{c}
-1, &0 \le x < 1 \\
2, &1 \le x \le 2
\end{array}
\right.
$$
Take... | Yes your proof looks rigorous and correct. You can also use the fact that a function with finitely many discontinuities in an interval satisfies that the integral of the function is equal to the sum of integrals of the function over the subintervals in-between the discontinuities, which is basically what you just prove... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Kind of basic combinatorical problems and (exponential) generating functions I have a pretty straightforward combinatorical problem which is an exercise to one paper about generating functions.
*
*How many ways are there to get a sum of 14 when 4 distinguishable dice are rolled?
So, one die has numbers 1..6 and a... | As ShreevatsaR pointed out it's sufficient to consider ordinary generating functions, since they already take into account that $3,4,3,4$ and $3,3,4,4$ are different. The first is coded as the coefficient of $x^3x^4x^3x^4$, while the second as the coefficient of $x^3x^3x^4x^4$ when considering the ogf $(x^1+\cdots+x^6)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/758950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Minimal polynomial: is $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$? I was wondering about the minimal polynomial of real number
$$u=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$$
over field $\mathbb{Q}$.
As you can see here, I worked out that $u$ is a root of monic rational polynomial $x^3+3x-4$. This is not irreducible... | Here's another way of looking at this through a reverse lens:
Let's solve $u^3+3u-4=0$ by Cardano's method, putting $u=x+y$.
Then $(x+y)^3-3xy(x+y)-(x^3+y^3)=0$ and we require:
$$x^3+y^3=4$$ and $$-3xy = 3 \text { so that }xy=x^3y^3=-1$$
Then we note that $x^3$ and $y^3$ are roots of the quadratic $$z^2-4z-1=0$$So that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Calculate the integral using Jacobian The problem asks you to calculate the following integral using Jacobian
$$\int{4x^2+y^2}dA$$ and it tells you to substitute $y+2x=u,y-2x=v,0<u<2,-2<v<0$, and the Jacobian is $\cfrac{1}{4}$.
Could anyone here help me out? Thank you.
| First, we determine $u$ and $v$ in term of $x$ and $y$. Using elimination/ substitution, it's easy to find that $x=\frac{1}{4}(u-v)$ and $y=\frac{1}{2}(u+v)$. Just adding and subtract $u$ and $v$. Then
\begin{align}
\int(4x^2+y^2)\,dA&=\int\int(4x^2+y^2)\,dx\,dy\\
&=\int_{-2}^0\int_0^2\left(4\left(\frac{1}{4}(u-v)\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/761637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Infinite Series $\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}$ I'm looking for a way to prove
$$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$
I know that
$$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{1}{4^{2m+1}}\left(\zeta\left(2m+1,\frac14\right)-\zeta\le... | Because it wasn't explicitly stated, I probably should state that $E_{2m}$ are the Euler numbers.
Let's integrate the function $$ f(z) = \frac{\pi \csc (\pi z)}{(2z+1)^{2m+1}}, \quad m \in \mathbb{N}_{\ge 0},$$
around a square contour with vertices at $\pm (N+\frac{1}{2}) \pm i(N+\frac{1}{2})$ , where $N$ is a positive... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/762813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 3
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Proving the area of an equilateral triangle How do you prove that How do you prove that for any equilateral triangle with side length s, area is $\frac{s^2 √3}{4}$ ? I tried using an equilateral triangle in a square, but I keep coming up with a $2x^2√3$ , as shown below. What am I doing wrong?
I started with the follow... | Let's consider a rectangle wit height $x\sqrt{3}$ and length $x$ such that the right angled triangle with legs as the sides forms half of the equilateral triangle (as shown in the picture in the question). Then, the area of the right angled triangle is $\dfrac{1}{2}x^2\sqrt{3}$. Two times this is the area of the equila... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/766640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the average value of $y=x^2\sqrt{x^3+1}$ on the interval $(0,2)$ I know the average value formula is $$\frac{1}{b-a}\int_a^b f(x)\;dx$$
I have no problem plugging in 0 and 2 for a and b respectively. I think i'm struggling actually taking the integral.
| here is an alternative method to substitution (actually its the same method, just arranged a bit differently).
$$\frac{1}{2-0}\int^2_0 x^2\sqrt{x^3+1}dx$$
$$=\frac{1}{2}\int^2_0 x^2\sqrt{x^3+1} dx\cdot\frac{d(x^3+1)/dx}{d(x^3+1)/dx}$$
$$=\frac{1}{2}\int^2_0 x^2\sqrt{x^3+1}\cdot\frac{d(x^3+1)}{3x^2}$$
$$=\frac{1}{6}\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/766756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find the minimum value of this expression with absolute values The expression is
$$|x-3| + |x-1| + |x| + |x+2| + |x+4|$$
I know that the minimum values for this expression is when x = 0 but is there any algebraic way to find this out? I did it on the calculator
| Let $f$ the function defined by the equation $f(x)=|x-3|+|x-1|+|x|+|x+2|+|x+4|$ for all $x\in \mathbb{R}$. $$x<-4\Longrightarrow f(x)=-(x-3)-(x-1)-x-(x+2)-(x+4)=-5x-2>18$$
$$-4\le x <-2 \Longrightarrow f(x)=-(x-3)-(x-1)-x-(x+2)+(x+4) = -3x+6>12$$
$$-2\le x <0 \Longrightarrow f(x)=-(x-3)-(x-1)-x+(x+2)+(x+4) = -x+10>10$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
How to solve this equation $2\cos(\frac {x^2+x}{6})=2^x+2^{-x}$ How do I solve for $x$ from this equation?
$$2\cos\frac {x^2+x}{6}=2^x+2^{-x}$$
| $-2\le2\cos(\frac{x^2 + x}{6})\le 2$ and $2\le2^x + 2^{-x}$.
Therefore the only possible solution is when both equal $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/770306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
using substitution wrongly Solving integral, first way:
$$\int \frac{du}{u^2-9}=-\int \frac{du}{9-u^2}$$
$$u={3\sin v}$$
$$du=3\cos vdv$$
$$-\int \frac{3\cos vdv }{9-9\sin^{2}v}=-\frac 13\int\frac{dv}{\cos v}=-\frac 13\ln\left(\sec v+\tan v\right)=-\frac13 \ln \frac {\frac u3}{\sqrt{1-\frac{u^2}{9}}}$$
$$-\frac13 \ln \... | The first term should be
$$
\begin{align}
-\frac 13\ln\left|\sec v+\tan v\right|+C&=-\frac 13\ln\left|\frac{3}{\sqrt{9-u^2}}+\frac{u}{\sqrt{9-u^2}}\right|+C\\
&=-\frac 13\ln\left|\frac{3+u}{\sqrt{9-u^2}}\right|+C\\
&=-\frac 13\ln\left|\sqrt{\frac{(3+u)(3+u)}{(3-u)(3+u)}}\right|+C\\
&=\frac 16\ln\left|\frac{3-u}{3+u}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/771413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Prove that $\sin\frac{\pi}{14}$ is a root of $8x^3 - 4x^2 - 4x + 1=0$ Prove that $\sin\frac{\pi}{14}$ is a root of $8x^3 - 4x^2 - 4x + 1=0$.
I have no clue how to proceed and tried to prove that the whole equation becomes $0$ when $\sin\frac{\pi}{14}$ is placed in place of $x$ but couldn't do anything further. I think... | Like my other answers $$\sin\frac\pi{14}=\cos\left(\frac\pi2-\frac\pi{14}\right)=\cos\frac{3\pi}7$$
Using this, $$\cos\frac{\pi}7-\cos\frac{2\pi}7+\cos\frac{3\pi}7=0$$
Now $\displaystyle\cos\frac{\pi}7=\cos\left(\pi-\frac{6\pi}7\right)=-\cos\left(2\cdot\frac{3\pi}7\right)$ (use Double Angle formula $\displaystyle\cos2A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/773131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Probability, Uniform Distribution. Suppose $A$ is a random number chosen uniformly from the interval $(-5,6)$,what is the probability that the quadratic equation $x^2 + Ax + 1 = 0$, has no real root?
This is my approach,
Using the quadratic equation we know,
$b^2 - 4ac < 0$ (has no real roots)
Therefore solving we get... | The first part looks fine - $x^2 + Ax + 1 = 0$ having no real solution means $(x + A/2)^2 = A^2/4 - 1$ < 0, which implies $|A| < 2$.
But then you're done! Since $A$ is uniformy distributed on $(-5,6)$, and since $(-2,2)$ lies fully within that interval, the probabily is simply the ratio of the interval lengths, i.e. $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/773358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Lagrange multiplier - space probe i am stuck on this question which uses the Lagrange multiplier. I am trying to construct the equations using the partial derivatives but the $x$'s and $y$'s cancel. can anyone help?
A space probe in the shape of the ellipsoid
$x^2 + y^2 + 3z^2 = 3$
enters a planet's atmosphere and begi... | Set $\Lambda \colon\mathbb R^4\to \mathbb R, (x,y,z,\lambda)\mapsto x^2+2y^2+6z+\lambda (x^2+y^2+3z^2-3)$.
Let $(x,y,z,\lambda)\in \mathbb R^4$.
The following holds:
$$\begin{cases} \Lambda _x(x,y,z,\lambda)&=2x+2\lambda x\\ \Lambda _y(x,y,z,\lambda)&=4y+2\lambda y\\ \Lambda_z(x,y,z, \lambda)&=6+6\lambda _z\\ \Lambda _... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/774068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Closed form of $ \int_0^{\pi/2}\ln\big[1-\cos^2 x(\sin^2\alpha-\sin^2\beta \sin^2 x)\big]dx$ Hello I am trying to solve an incredible integral given by
$$
\int_0^{\pi/2}\ln\big[1-\cos^2 x(\sin^2\alpha-\sin^2\beta \sin^2 x)\big]dx=\pi \ln\bigg[\frac{1}{2}\left(\cos^2\alpha +\sqrt{\cos^4 \alpha +\cos^2\frac{\beta}{2} \si... | With the shorthands $r=\frac{\sin^2\alpha-\sin^2\beta}{2\sin\beta}$, $s=\frac{\sin^2\alpha+\sin^2\beta}{2\sin\beta}$ and the known integral
$$\int_{0}^{\pi/2} \ln(p + q \cos^2x)dx
= \pi\ln\frac{\sqrt p+\sqrt{p+q}}2
$$evaluate
\begin{align}
I=& \int_0^{\pi/2}\ln\left[1-\cos^2 x(\sin^2\alpha-\sin^2\beta \sin^2 x)\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/775200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Find the remainder when $15!$ is divided by $31$. Find the remainder when $15!$ is divided by $31$. I know I have to apply Wilsons theorem but i am a little confused how.
| This might be hitting a fly with a brick but.....
Primes less than or equal to 15: 2,3,5,7,11,13.
How many multiples of 2,4,8 are less than or equal to 15:7,3,1
How many multiples of 3,9:5,1
How many multiples of 5: 3
How many multiples of 7: 2.
11,13 > 15/2.
So $15! = 2^{7+3+1}3^{5+1}5^37^211\cdot13=2^{11}3^65^27^211... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove or disprove inequality $a^2+b^2+c^2\ge a^rb^{2-r}+b^rc^{2-r}+c^ra^{2-r}$. If $a$, $b$ and $c$ are real numbers greater than $0$ and $r$ is a real number with $0 \le r \le 2$. Does inequality $$a^2+b^2+c^2\ge a^rb^{2-r}+b^rc^{2-r}+c^ra^{2-r}$$ hold?
| Hint: Use Rearrangement Inequality, noting $a^r, b^r, c^r$ and $a^{2-r}, b^{2-r}, c^{2-r}$ are similarly ordered.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/781108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Factor a quadratic equation to get two binomials I'm wrestling with this quadratic and trying to figure out how to factor it:
$$3x^2 - 5x + 2 = 0$$
I know that the product of the last terms of the binomial for an equation equals the third term of the polynomial. Also, the sum of the products of those two numbers should... | Let me offer a different approach. Try to complete the square, by adding and subtracting suitable terms. $3x^2$ begs for an extra $x^2$ to be added to it, so there we go:
$$
\begin{aligned}
3x^2−5x+2 &= 4x^2 -4x + 1 - x^2 - x + 1 \\
&= (2x-1)^2 -(x^2- 2x+1) - 3x + 2 \\
&= (2x-1)^2 -(x-1)^2 - (3x - 2) \quad \color{red}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/782816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ ... Problem : Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ and $f(0) =2+\sqrt{3}+\sqrt{5}$ Prove that the equation $f(2009)x^2 +2f(0)x +... | A continuous function only taking irrational values is constant, hence $f(2009)x^2 + 2f(0)x + f(2009)=0$ simplifies to $x^2+2x+1=0$ or $(x+1)^2=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/783130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\lim_{n\to\infty} (\sqrt{n^2+n}-n) = \frac{1}{2}$ Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$
Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it sp... | You can simply solve the inequality $\frac{1}{\sqrt{1+\frac{1}{n}}+1}>\frac{1-2\epsilon}{2}$and find pick N = $\frac{(1-2\epsilon)}{8\epsilon^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/783536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 10,
"answer_id": 2
} |
determining the limit tending to infinity of n times squared roots I'm having troubles showing that
$$
\lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+.....+\sqrt{n}-\frac23n\sqrt{n}}{\sqrt{n}} = \frac12.
$$
I have tried this but I am not able to solve this.
| Applied Stolz-Cesaro Theorem:
$$
\lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+.....+\sqrt{n}-\frac23n\sqrt{n}}{\sqrt{n}} = \lim_{n\rightarrow \infty}\frac{\sqrt{n+1}-\frac{2}{3}(n+1)\sqrt{n+1}+\frac{2}{3}n\sqrt{n}}{\sqrt{n+1}-\sqrt{n}}=\frac{1}{3}\lim_{n\rightarrow \infty}\frac{2n\sqrt{n}-(2n-1)\sqrt{n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/784653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Understanding substitution to compute primitive To compute the primitive
$$\int x^2 \sqrt[3]{x^3+3}\ dx$$
I am trying this:
$t = x^3+3$
$dt = 3x^2dx$
but
$\int x^2 \sqrt[3]{x^3+3}\ dx=\ ?$
How to continue from here?
| Make the substitution $t=x^3+3$. Then $dt=3x^2dx$, and we have
$$
\frac{1}{3}\int t^{1/3}dt = \frac{1}{3}\left(\frac{3}{4}t^{4/3}\right) + C=\frac{1}{4}\left( x^3+3\right)^{4/3} +C.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/790405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Find the axes of the ellipse What are the equations of the major and minor axes of the ellipse $x^2+2y^2-2xy-1=0$. The centre of the ellipse is $(0,0)$ but the axes are tilted (with respect to $x-y$ axes). I don't know how to find those.
| You take an ellipse of the form $a x^2 + b y^2 + c x y + d x + e y + f = 0$ and match coefficients with the equation of an ellipse centered about $(x_c,y_c)$ at an oblique angle $\theta$
$$ \left( \frac{ (x-x_c) \cos\theta + (y-y_c) \sin\theta}{R_x} \right)^2
+ \left( \frac{ -(x-x_c) \sin\theta + (y-y_c) \cos\theta}{R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Let $z=\ln \tan\frac xy.$ What is $z_x$ and what is $z_y$? Let $$z=\ln \tan\frac xy.$$ What is $z_x$ and what is $z_y$?
Thanks ahead:)
What I have tried:
$$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$
$$z_y=\frac {-x}{\tan \frac xy (x^2+y^2)}.$$
I am not sure I am r... | As pointed out in comments, you misread $\tan$ as $\tan^{-1}$.
The answers are:
$$
\begin{array}{c}
z_x = \frac{1}{\tan\frac{x}{y}}\frac{1}{y}\sec^2\frac{x}{y}=\frac{\sec\frac{x}{y}\csc\frac{x}{y}}{y} \\
z_y = -\frac{1}{\tan\frac{x}{y}}\frac{x}{y^2}\sec^2\frac{x}{y}=\frac{-x \sec\frac{x}{y}\csc\frac{x}{y}}{y^2}
\end{ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/791045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integral $\int_0^1 \log x \frac{(1+x^2)x^{c-2}}{1-x^{2c}}dx=-\left(\frac{\pi}{2c}\right)^2\sec ^2 \frac{\pi}{2c}$ Hi I am trying to prove this result $$
I:=\int_0^1 \log x \frac{(1+x^2)x^{c-2}}{1-x^{2c}}dx=-\left(\frac{\pi}{2c}\right)^2\sec ^2 \frac{\pi}{2c},\quad c>1.
$$
Thanks. Since $x\in[0,1]
$ we can write$$
I=\s... | Another derivation that uses only very elementary complex analysis and no special functions:
(all integrals are to be regarded as Cauchy PV integrals if necessary)
By substituting $x\mapsto 1/x$, we find that the original integral equals
$$\int_1^{\infty} (-\log x) \frac{(1+1/x^2)x^{2-c}}{1-x^{-2c}}\frac{dx}{x^2} = \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/791870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Beautiful Indefinite Integrals. These are some of the integrals with beautiful solutions I came across-
$$\int \frac{x^2}{(x\sin x+\cos x)^2} dx$$
$$\int\frac {1}{\sin^3x+\cos^3x} dx$$
$$\int \frac{1}{x^4+1}dx$$
I'd love if you share some of the ones you came across.
| \begin{align}
I_1 & = \int \sqrt{ \sqrt{ x + 2\sqrt{2x-4} } +
\sqrt{ x - 2\sqrt{2x-4} } } \,\mathrm{d}x \, , \quad x>4\\
I_2 & = \int \log( \log x) + \frac{2}{\log x} - \frac{1}{(\log x)^2} \mathrm{d}x \\
I_4 & = \int (1 + 2x^2) e^{x^2}\, \mathrm{d}x \\
I_5 & = \int \frac{\sqrt{x+\sqrt{x^2+1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/796166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 1
} |
Partial fraction integral Question:
$\int \dfrac{5 }{(x+1) (x^2 + 4) } dx $
Thought process:
I'm treating it as a partial fraction since it certainly looks like one.
I cannot seem to solve it besides looking at it in the "partial fraction" way.
My work:
1) Focus on the fraction part first ignoring the $\int $ and $d... | Your equation is incorrect--you have some algebra mistakes that are causing you to not get a solution (which is obvious unless you start plugging in values to $x$):
$$
\frac{5}{(x + 1)(x^2 + 4)} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 4}
$$
Now cross multiply to get the numerator:
\begin{align}
5 =& A(x^2 + 4) + (Bx + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find the two points where the shortest distance occurs on two lines Find the point P on $\vec{AB}$ and point Q on $\vec{CD}$ such that $\vec{PQ}$ is the shortest distance
between the lines AB and CD, given $\vec{AB} = \begin{pmatrix}
1\\
0\\
2\\
\end{pmatrix}
+ u\begin{pmatrix}
-2\\
2\\
1\\
\end{pmatrix}
,\vec{C... | If the normal vector is not provided in the question, you can still solve the question.
Any point $P$ on line $\vec{AB}$ given $u$ can be represented as
$\begin{pmatrix}
1-2u \\
2u\\
2+u
\end{pmatrix}$
, whilst point any point $Q$ on line $\vec{CD}$ given $v$ can be represented as
$\begin{pmatrix}
2v \\
1-v\\
1-2v
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$ I know that $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$ does not exist, because the lateral limits are different and I also know that the absolute value on the denominator has something to do with it. But I can´t get my mind a... | The key is to realize that absolute value is a piecewise defined function in disguise. Recall that $|u| = u$ if $u > 0$ and $|u| = -u$ if $u < 0.$ In your case, $u = 2x – 3,$ so $u > 0$ corresponds to $x > \frac{3}{2}$ (solve $2x – 3 > 0)$ and $u < 0$ corresponds to $x < \frac{3}{2}$ (solve $2x – 3 < 0).$
Thus, we want... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
log and poisson-like integral Here is a fun looking one some may enjoy.
Show that:
$$\int_{0}^{1}\log\left(\frac{x^{2}+2x\cos(a)+1}{x^{2}-2x\cos(a)+1}\right)\cdot \frac{1}{x}dx=\frac{\pi^{2}}{2}-\pi a$$
| Starting from
$$
{\rm Log}(1+x e^{ia})=\sum_{n=1}^\infty\frac{(-1)^{n-1}e^{ina}}{n}x^n
$$
we see that
$$
\int_0^1{\rm Log}(1+x e^{ia})\cdot \frac{1}{x}dx=\sum_{n=1}^\infty\frac{(-1)^{n-1}e^{ina}}{n^2}
$$
Taking real parts we get
$$
\int_0^1 \log|1+x e^{ia}|\cdot \frac{1}{x}dx=\sum_{n=1}^\infty\frac{(-1)^{n-1}\cos(na)}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/808144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Any purely geometric solution to this problem?
What is the largest possible area of a rectangle(in square units) inscribed in the triangle shown in the picture above?
| Let $\triangle ABC$ be the given triangle with $AB = 10$, $AC = 17$, and $BC = 21$. Choose an arbitrary point $M$ on the side $AB$, and let $x = AM$. Let $N$ be the point on $AC$ such that $MN \parallel BC$, and points $P$, and $Q$ on $BC$ such that $NP \perp BC$, and $MQ \perp BC$. Thus $MNPQ$ is a rectangle. Let $h_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/812282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Showing that $3x^2+2x\sin(x) + x^2\cos(x) > 0$ for all $x\neq 0$ I got this question:
Show that for all $x\neq 0$, $3x^2+2x\sin(x) + x^2\cos(x) > 0$
I tried to show it but got stuck.
| Another approach:
$$
3x^2+2xsin(x)+x^2cos(x) = x(3x+\sqrt{2+x^2}sin(\alpha+x))
$$
where, $\alpha = tan^{-1}(x/2)$
Since, $ x> 0$, we need only prove: $3x-\sqrt{2+x^2}>0$, for $x>0$
or $3x>\sqrt{2+x^2}$, squaring both sides, and rearranging terms, $x>\frac{1}{2}$
Also, for $0<x<\frac{1}{2}<\frac{\pi}{2}$, all terms in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/812453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How find the prime $p$ such $p\mid\sum_{k=1}^{p+2}\frac{T_{k}}{k+1}$
let $k\in N^{+}$, and such
$$T_{k}=\sum_{i=1}^{k}\dfrac{1}{i\cdot 2^i}$$
Find all prime number $p$, such that
$$p\mid\sum_{k=1}^{p+2}\dfrac{T_{k}}{k+1}$$
I think this problem maybe use this Abel transformation:see http://en.wikipedia.org/wiki... | What I get from partial summation is:
$$\begin{eqnarray*}I_p=\sum_{k=1}^{p+2}\frac{1}{k+1}\sum_{i=1}^{k}\frac{1}{i\,2^i}&=&(H_{p+3}-1)T_{p+2}-\sum_{k=1}^{p+1}\frac{H_{k+1}-1}{(k+1)2^{k+1}}\\&=&H_{p+3}T_{p+2}-\sum_{k=1}^{p+2}\frac{H_k}{k\,2^k}.\tag{1}\end{eqnarray*}$$
Now, it is well-known that $H_{p-1}\equiv 0\pmod{p^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/813438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find volume of sphere $ x^2+y^2+z^2=9$ bounded by planes $z=0$ and $z=2$ using double integral Find volume of sphere $x^2+y^2+z^2=9 $ bounded by planes $z=0$ and $z=2$ using double integral
I tried to take the total volume of the bigger hemisphere but i get zero, i managed to take the volume of the smaller hemisphere o... | We need to use the fact that $x^2+y^2=r^2$, so we can convert to polar coordinates. That is, $x^2+y^2+z^2=9$ implies $z = \pm \sqrt{9-(x^2+y^2)}=\pm\sqrt{9-r^2}$.
However, since the volume of the solid we want is above $z=0$, we only need to consider the top half of the sphere, that is, $z=+\sqrt{9-r^2}$.
Now here's ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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} |
A system of equations Given three equations $x^2+y^2+xy=a$, $y^2+z^2+yz=b$ and $x^2+z^2+xz=c$, how can I solve for $x,y$ and $z$ in terms of $a,b$ and $c$?
| If we multiply second equation by $-1$ and add to first we have:
$x^2+y^2+xy-y^2-z^2-yz=(x+y+z)(x-z)=a-b$
Next do the same with other equation:
$(x+y+z)(y-z)=a-c$
$(x+y+z)(y-x)=b-c$
Now if $a \neq c$ we can divide first by second and get:
$\frac{x-z}{y-z}=\frac{a-b}{a-c}$, so:
$(x-z)(a-c)=(a-b)(y-z)$. It's linear.
Next... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How prove this interesting identity $(y_{1})^2\cdot y_{2}\cdot y_{3}=x^2_{1}\cdot x_{2}\cdot x_{3}$
let $0<x_{1}<x_{2}<x_{3}$, and there exsit $a$ such
$$\begin{cases}
y_{1}=x_{1}-\ln{x_{1}}=\dfrac{x^2_{1}}{ax_{1}+\ln{x_{1}}}\\
y_{2}=x_{2}-\ln{x_{2}}=\dfrac{x^2_{2}}{ax_{2}+\ln{x_{2}}}\\
y_{3}=x_{3}-\ln{x_{3}}=\dfrac... | Note $x_i>0$, and since $e^x \geq 1+x$ we have $y_i=x_i-\ln x_i \geq 1$. We have
$$y_i=x_i-\ln x_i=\frac{x_i^2}{ax_i+\ln x_i}=\frac{x_i^2}{ax_i+x_i-y_i}=\frac{x_i^2}{(a+1)x_i-y_i}$$
$$(a+1)x_iy_i-y_i^2=x_i^2$$
Let $a+1=2b$, so $2bx_iy_i-y_i^2=x_i^2$.
$$(x_i-by_i)^2=(b^2-1)y_i^2$$
$$x_i=\left(b \pm \sqrt{b^2-1}\right)y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Determinant of specially structured block matrix How do you compute the determinant of the following $nd \times nd$ block matrix?
$$M = \begin{bmatrix}A+B & A & A & \dots & A & A\\ A & A+B & A & \dots & A & A\\ A & A & A+B & \dots & A & A\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ A & A & A & \dots & A+B &... | Let us assume that $B$ is invertible. Write
$$\begin{array}{rl} M &= \begin{bmatrix} A + B & A & \ldots & A\\ A & A + B & \ldots & A\\ \vdots & \vdots & \ddots & \vdots\\A & A & \ldots & A + B \end{bmatrix}\\\\ &= \begin{bmatrix} B & O_d & \ldots & O_d\\ O_d & B & \ldots & O_d\\ \vdots & \vdots & \ddots & \vdots\\ O_d ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Prove that for $n$ and $m$ integers: $ 3^mn \mid \sum\limits_{k=0}^{m} {\binom{3m}{3k}}(3n-1)^k$ If $m$ and $n$ are positive integers, with $m$ odd, then prove that:
$$3^mn \mid \sum\limits_{k=0}^m \binom{3m}{3k} (3n-1)^k$$
Proving divisibility by $3n$, we look at $\sum\limits_{k=0}^m (-1)^k\binom{3m}{3k}$. My idea is... | Let us denote
$$A_m = \sum_{k=0}^m \binom{3m}{3k}(3n-1)^k.$$
Writing $z = \sqrt[3]{3n-1}$ and $\rho = e^{2\pi i/3}$, we find
$$3A_m = (1+z)^{3m} + (1+\rho z)^{3m} + (1+ \rho^2 z)^{3m}.$$
The sequence given by
$$3u_k = (1+z)^k + (1+\rho z)^k + (1+\rho^2 z)^k$$
satisfies the linear recurrence
$$u_{k+3} = 3 u_{k+2} - 3 u_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Integrating $(3x + 1) / x^\frac{1}{2}$ I am trying to integrate the following equation, but my answer is different from the textbook and I cannot see where I am going wrong:
\begin{align} \int_1^2\frac{ 3x + 1}{x^{1/2}}dx
&= \int_1^2 \frac{3x}{x^{1/2}}dx + \int_1^2 \frac{1}{x^{1/2}}dx \\
&= \int_1^2 3x^{1/2}dx + \i... | $$\begin{align} \int_1^2 3x^{1/2} \,dx + \int_1^2 x^{-1/2}\,dx & = \frac{3x^{3/2}}{{\frac 32}} + \frac{x^{1/2}}{1/2}\Big|_1^2\\ \\
& = 2 x^{3/2} + 2x^{1/2}\Big|_1^2 \\ \\
& = 4\sqrt 2 + 2\sqrt 2 - (2 + 2)\\ \\ & = 6\sqrt 2 - 4\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/823037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Product of Gamma functions II What is the value of the product of Gamma functions
\begin{align}
\prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right)
\end{align}
and can it be shown that
\begin{align}
\prod_{k=1}^{20} \Gamma\left(\frac{k}{10}\right) \approx \frac{\pi^{9}}{54}
\end{align}
and
\begin{align}
\prod_{k=1}^{40} \... | Since @robjohn has done an excellent job at providing an answer in detail I will add my comments as an additional solution. The results presented here follow the numbering in robjohn's work.
In robjohn's equation (12) the factor
\begin{align}
\frac{9! \ \pi^{9}}{2 \cdot 5^{10}}
\end{align}
has been provided. Numerical... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Evaluate the sum $\sum^{\infty}_{n=1} \frac{n^2}{6^n}$ Evaluate the sum $\sum^{\infty}_{n=1} \frac{n^2}{6^n}$
My approach :
$= \frac{1}{6}+\frac{2^2}{6^2}+\frac{3^2}{6^3} +\cdots \infty$
Now how to solve this I am not getting any clue on this please help thanks.
| Break $n^2$ into two parts: $\underbrace{n(n-1)} + \underbrace{{}\quad n\quad {}}$.
The first part appears in the second derivative of $x^n$ and the second part in the first derivative:
\begin{align}
\sum_{n=1}^\infty n^2 x^n & = \sum_{n=1}^\infty x^2\Big(n(n-1) x^{n-2} \Big) + x\Big(nx^{n-1}\Big) \\[10pt]
& = \left(x^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is this fourier even? $$
f(x) =
\begin{cases}
\frac{\pi}{4}-\frac{x}{2} & [0,\pi] \\
-\frac{3\pi}{4}+\frac{x}{2}, & (\pi,2\pi)
\end{cases}
$$
Is it right to compute only $a_n \text{ and } a_0$ coefficient for fourier series because $f(x)$ is even for fourier? How can I proove it since $f(x)!=f(-x)$
| For $2\pi$-periodic function $f(x)$ defined as:
$$f(x) =
\begin{cases}
\frac{\pi}{4}-\frac{x}{2} & x\in [0,\pi] \\
-\frac{3\pi}{4}+\frac{x}{2}, & x\in(\pi,2\pi)
\end{cases}$$
We can calculate
$$f(2\pi-x) =
\begin{cases}
\frac{\pi}{4}-\frac{2\pi-x}{2} & x\in [\pi,2\pi] \\
-\frac{3\pi}{4}+\frac{2\pi-x}{2}, & x\in (0,\pi)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/830401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How would I factor $a^3+b^3+c^3-6abc$ How would I factor the polynomial $a^3+b^3+c^3-6abc$? The values are homogenous, so so must be the factors. I don't know where to go from there.
| $$a^3+b^3+c^3-6abc= (a+b)^3-3ab(a+b)+c^3-6abc
\\(a+b+c)^3-3(a+b)c(a+b+c) -3ab(a+b)-6abc
\\=(a+b+c)^3 -3(a+b)c(a+b+c)-3ab(a+b)-(3+3)abc
\\=(a+b+c)^3-3c((a+b)(a+b+c)) -3ab(a+b+c) $$
now factor $$ (a+b+c) $$ and go on
$$(a+b+c)((a+b+c)^2 -3c(a+b) -3ab) )
\\=(a+b+c)(a^2+b^2+c^2+2ab+2ac+2bc-3ab-3bc-3ac)
\\=(a+b+c)(a^2+b^2+c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/831254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Prove this inequality: If $|x+3|< 0.5$, show that $|4x+13| < 3$ If $|x + 3| < 0.5$, show that $|4x + 13| < 3$
This is what I've got so far:
$|4x + 13| = |(x + 3) + (3x + 10)|$
by the Triangle Inequality:
$|(x + 3) + (3x + 10)| \le |x + 3| + |3x + 10|$
Now I continue to apply the Triangle Inequality to reach:
$|(x + 3... | You can also remove the absolute value and work with a two-sided equation. It's a little simpler, as it doesn't involve the triangle inequality.
$$-0.5<x+3<0.5$$
$$-2<4(x+3)<2$$
$$-2<4x+12<2$$
$$-1<4x+13<3$$
Therefore:
$$|4x+13|<3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/832063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Find the coefficients $a, b, c$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation.
Find the coefficients $a, b, c$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation $y = ax^3 + bx^2 + cx + d$.
I have no clue how to solve this. This looks like not... | $\begin{vmatrix}
x^3&x^2&x&1&y\\
0^3&0^2&0&1&10\\
1^3&1^2&1&1&7\\
3^3&3^2&3&1&-11\\
4^3&4^2&4&1&-14\\
\end{vmatrix}=
\begin{vmatrix}
x^3&x^2&x&1&y\\
0&0&0&1&10\\
1&1&1&1&7\\
27&9&3&1&-11\\
64&16&4&1&-14\\
\end{vmatrix}=
-72x^3+432x^2-144x-720+72y=
x^3-6x^2+2x+10-y=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/833903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Is there a simpler/better proof of this simple trigonometric property? The sine function has the following nice property : for any
$x,y$, we have $\sin(x)+\sin(y)=\sin(x+y)$ iff at least one of
$x,y,x+y$ is $0$ modulo $2\pi$.
I sketch below my current proof of it, which I find somewhat
unsatisfying. Does anyone know a... | We have $\DeclareMathOperator{\Ima}{Im}$
$$\begin{align}
\sin (x+y) - \sin x - \sin y &= \Ima \left(e^{i(x+y)} - e^{ix} - e^{iy}\right)\\
&= \Ima \left(e^{i(x+y)} - e^{ix} - e^{iy} + 1\right)\\
&= \Ima \left(\left(e^{ix}-1\right)\left(e^{iy}-1\right)\right)\\
&= \Ima \left(e^{i(x+y)/2}\left(e^{ix/2} - e^{-ix/2}\right)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/834321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
How to prove inequality $\frac{a}{a+bc}+\frac{b}{b+cd}+\frac{c}{c+da}+\frac{d}{d+ab}\ge 2$ Question:
Let $$a,b,c,d>0,a+b+c+d=4$$
show that
$$\dfrac{a}{a+bc}+\dfrac{b}{b+cd}+\dfrac{c}{c+da}+\dfrac{d}{d+ab}\ge 2$$
when I solved this problem, I have see following three variables inequality:
Assumming that $a,b,c>0,a+b+c... | Let's denote the left-hand expression by L.
Then, by Cauchy-Schwarz:
$[a(a + bc) + b(b + cd) + c(c + da) + d(d+ab)]L \geq (a + b + c + d)^2 = 16$
Now, we have to show
$[a(a + bc) + b(b + cd) + c(c + da) + d(d+ab)] \leq 8$ and we are over.
Let's write it down like:
$A(a, b, c, d) = a^2 + b^2 + c^2 + d^2 + ac(b+d) + bd(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/837565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
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Equation $1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$ How to prove that the following equation: $$1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$$
has for solution(in real numbers): $|x|=|y|=1~$ only.
Any hint would be appreciated.
| Note that the proposed equation is equivalent to
$$
(1-x^2y^4)^2+(1-x^4y^2)^2+x^4y^4(x^2-y^2)^2=0
$$
So, any solution satisfies $x^2y^4=x^4y^2=1$ and $xy(x^2-y^2)=0$. This implies clearly that $x^2=y^2=1$.$\qquad\square$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/838838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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$a,b \in \mathbb N $ , $b$ odd $\implies$ $ \dfrac{2a^2-1}{b^2+2} \notin \mathbb Z $? If $a,b$ are positive integers and $b$ is odd , then is it ever possible that $ \dfrac{2a^2-1}{b^2+2} $ is an integer ?
| Since $b$ is odd, $b^2 + 2\equiv 3\pmod{8}$. Thus $2$ is not a quadratic residue mod $(b^2 + 2)$. Hence $2a^2\not\equiv 1\pmod{b^2 + 2}$, from which the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/839241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$? I have no idea how to start, it looks like integration by parts won't work.
$$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$
If someone could shed some light on this I'd be very thankful.
| I try to tackle the problem by differentiation under integral sign.
Using the famous result:
$$
\int_0^{\infty} \frac{\cos a x}{x^2+t^2} dx=\frac{\pi}{2 t} e^{-a t}
$$
where $a,t>0.$
we get the result of the integral
$$
\int_0^{\infty} \frac{\cos a x}{1+t x^2} d x=\frac{\pi}{2} \frac{1}{\sqrt{t} e^{\frac{a}{\sqrt t}}}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/839426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$ Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$
I went wrong somewhere, this is what I have so far (this is in polar):
$z=4\left(\cos\left(\frac{11\pi}{6}\right)+\sin\left(\frac{11\pi}{6}\right)\right) $
$w=\sqrt2\left(\cos\left(\frac{7\p... | First, notice that the argument of $w$ is $\frac{3 \pi}{4}$, not $\frac{7\pi}{4}$. And you forgot to put the "$i$"'s together with the sines. Just a little distraction. The other argument, and the absolute values are ok. You had setup everything else correctly. Using the right value above, we get: $$\begin{align}zw &=4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/845086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Derivative of $\frac{1}{2s}-\frac{5}{4s^{3}}$
$r=\dfrac{1}{2s}-\dfrac{5}{4s^3}$
$r=\dfrac{1}{2}s^{-1}-\dfrac{5}{4}s^{-3}$
$r^{\prime}=-\dfrac{1}{2}s^{-2}-\dfrac{5}{4}(-3)s^{-4}$
$r^{\prime}=-\dfrac{1}{2s^{2}}+\dfrac{15}{4s^{4}}$
Is this correct? Because I solved it again a different way by using the quotient rule a... | Finding a common denominator and using the quotient rule we have
$$r=\frac{1}{2s}-\frac{5}{4s^3}$$
$$r=\frac{2s^2-5}{4s^3}$$
$$r'=\frac{4s^3\cdot4s-(2s^2-5)\cdot12s^2}{16s^6}=\frac{16s^4-(24s^4-60s^2)}{16s^6}=\frac{60s^2-8s^4}{16s^6}$$
$$r'=-\frac{1}{2s^2}+\frac{15}{4s^4}$$
As you calculated.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/847534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Fibonacci Calculation using a larger matrix So the formula to generate the fibonacci sequence in matrix form is:
$$
\begin{pmatrix}
1 & 1 \\
1 & 0 \\
\end{pmatrix}^n =
\begin{pmatrix}
F_{n+1} & F_n \\
F_n & F_{n-1} \\
\end{pmatrix}
$$
So, is there a way to generalize it to a larger matrix, a 4x4 for ... | $$
\begin{pmatrix}
1 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 2 & 1 \\
0 & 0 & 1 & 1 \\
\end{pmatrix}^n =
\begin{pmatrix}
F_{n+1} & F_n & 0 & 0 \\
F_n & F_{n-1} & 0 & 0 \\
0 & 0 & F_{2n+1} & F_{2n} \\
0 & 0 & F_{2n} & F_{2n-1} \\
\end{pmatrix}
$$
The point is you're asking for something ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/849926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Express $\log_5 288$ in terms of decimal logarithms $\log 2$ and $\log 3$ Assuming $a=\log 2$ and $b=\log 3$ (log is the base 10 logarithm). I have to find $\log_5 288$. How can I do this?
I've tried transforming $\log2$ to $\frac{\log_5 2}{\log_5 10}$ and same for $b$. Then it's $$\frac{5\log_5 2+2\log_5 3}{\log_5 10... | \begin{align}
\log_5 288 & = \frac{\log_{10} 288}{\log_{10} 5} = \frac{\log_{10}(2\cdot2\cdot2\cdot2\cdot2\cdot3\cdot3)}{\log_{10} (10/2)} \\[15pt]
& = \frac{\log_{10}2 + \log_{10}2 + \log_{10}2 + \log_{10}2 + \log_{10}2 + \log_{10}3 + \log_{10}3 + }{\log_{10} 10 - \log_{10} 2} \\[15pt]
& = \frac{\log_{10}2 + \log_{10... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/850284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two diophantine equations with lots of unknowns Is it possible (tractable) to determine if the following system of equations has any nontrivial solutions (ie, none of the unknowns are zero) in the domain of integers?
$$A^2 + B^2=C^2 D^2$$
$$2 C^4 + 2 D^4 = E^2 + F^2$$
| for the second one, take $C > D > 0,$ then
$$ E = C^2 - D^2, \; \; \; F = C^2 + D^2 $$
If you wanted a system, take any $C,D \equiv 1 \pmod 4$ distinct primes, such as $5,13.$ We get the Pythagorean triple $16^2 + 63^2 = 65^2 = 5^2 13^2.$ Then $2 \cdot 5^4 + 2 \cdot 13^4 = (13^2 - 5^2)^2 + (13^2 + 5^2)^2 = 144^2 + 194^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/850631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Generating function satisfying a second degree equation I got this problem in an exercise list:
Let $G(x)$ be the generating function of the numeric sequence $(C_n; n \geq 0)$ satisfying the recurrence equation:
$$C_n = \sum_{k=0}^{n-1}C_kC_{n-k-1}, C_0=C_1=1.$$
Show that $xG(x)^2 - G(x)+1=0$ and conclude that
$$G(x)=... | You can tell that the minus sign is correct, since you know that $G(x)$ is a power series, and if you used the plus sign, the numerator would start $1+1+...$, so that $G(x)$ would have a term $\frac{1}{x}$. To see that $G(x) = \sum_{n=0}^{\infty}\frac{1}{n+1}\binom{2n}{n}x^n$, note that
$$(1+x)^{1/2} = \sum_0^\infty \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/853892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Orthogonal Projection of $v$ on sub-space $U$
What is the orthogonal projection of $v = (1,5,-10)$ on the sub-space $U = Sp\{(5,-2,1),(1,2,-1)\}$?
Well, I managed to compute it on the way I've been taught by the book, but it doesn't seem to work. Any chance you guys can solve it? If needed, I can provide the way I tr... | Hint:
1) Compute an orthonormal base $u_1,u_2$ of $U$ using Gramm-SChmidt
2) Consider the projector $p_U(x) = \sum_{i=1}^2 <u_i,x>u_i$
3) Compute $p_U(v)$
So let's do it:
Note that $$\langle \begin{pmatrix} 1\\ 2 \\ -1 \end{pmatrix}, \begin{pmatrix}5\\ -2 \\ 1 \end{pmatrix}\rangle = 0 \quad \text{ and }\quad \|\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/856246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Convergence N'th Harmonic number minus the Natural Logarithm of N. I was hoping if someone could show me the proof of exactly why this converges to the Euler–Mascheroni constant.
| The sequence,
$$1+\frac{1}{2}+\cdots +\frac{1}{n}- \ln n $$ converges.
Consider the series
$$\sum\limits_{n=1}^{\infty} \frac{1}{n}-\ln\left( 1+\frac{1}{n} \right).$$
We show that this series converges. We use the inequality,
$$\ln (1+x) < x \qquad \text{for} -1<x <\infty.$$
First, all of its terms are positive, since
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/857473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How find this integral $I(a,b)=\iiint_{x^2+y^2+z^2\le 1}(ax+by)^2\,dx\,dy\,dz$ Find this integral
$$I(a,b)=\iiint_{x^2+y^2+z^2\le 1}(ax+by)^2\,dx\,dy\,dz$$
since
$$(ax+by)^2=a^2x^2+b^2y^2+2abxy$$
so
$$I=I_{1}+I_{2}+I_{3}$$
where
$$I_{1}=\iiint_{x^2+y^2+z^2\le 1}ax^2\,dV=a\iiint_{x^2+y^2+z^2\le 1}x^2\,dV$$
since
$$\iiin... | Expanding out, your integral is
$$a^2 \iiint_{x^2+y^2+z^2\le 1}x^2\,dV - 2ab \iiint_{x^2+y^2+z^2\le 1}xy\,dV + b^2 \iiint_{x^2+y^2+z^2\le 1}y^2\,dV$$
The middle term is zero since for fixed $y$ and $z$ the integrand is odd in $x$ and the domain of integration is a symmetric interval in $x$. Furthermore, by symmetry th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/858157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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 $n$?
| $\Large \frac{2^{n+1}}{n+1}=\frac{4+2^{n}}{3}$
this equation tells us that $n+1>0 \Rightarrow n>-1 $((i)we can't divide by zero (ii) RHS is +ve so LHS must also be +ve)
$\Large \frac{2^{n+1}}{n+1}=\frac{8+2^{n+1}}{6}=\frac{8}{6}+\frac{2^{n+1}}{6}$(multiply and divide RHS by $2$)
$\Large \frac{2^{n+1}}{n+1}-\frac{2^{n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/859805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solve the System of Equations in Real $x$,$y$ and $z$ Solve for $x$,$y$ and $z$ $\in $ $\mathbb{R}$ if
$$\begin{align} x^2+x-1=y \\
y^2+y-1=z\\
z^2+z-1=x \end{align}$$
My Try:
if $x=y=z$ then the two triplets $(1,1,1)$ and $(-1,-1,-1)$ are the Solutions.
if $x \ne y \ne z$ Then we have
$$\begin{alig... | No solutions exist for real, distinct $(x,y,z)$.
Writing $f(x)=x^2+x-1$, the existence of such a solution would mean that $f$ has a real point of least period 3; that is, $f^3(x)=x$ for some real $x$ with $f(x)\neq1$ i.e. $x\neq \pm 1$. Sarkovsky's theorem then implies that $f$ has points of arbitrary least period. In ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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If $a + b + c = 0$ prove that If $a + b + c = 0$, prove that
1)$$
\sum_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 3
$$
2)$$
\prod_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 1
$$
There is a solution that uses two cubic equations. First is of form $x^3+px-q=0$ that has the roots $a,b$ and $c$. After he is forming anoth... | From $\displaystyle x^3+px-q=0, abc=q, ab+bc+ca=p$
and $\displaystyle a^3+pa-q=0\ \ \ \ (1)\iff a^3=q-pa\ \ \ \ (2)$
Assuming $q\ne0,$
$\displaystyle y=\frac{4bc-a^2}{bc+2a^2}=\frac{4abc-a^3}{abc+2a^3}$
$\displaystyle y=\frac{4q-a^3}{q+2a^3}$
Express $a^3$ in terms of $y$ and compare the value of $a^3$ with that of $(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to express a trigonometic equation in $\sin 2\theta $ and $\cos 2\theta $? How do I express the given equation in $\sin 2\theta $ and $\cos 2\theta $ in terms of x?
$x + 3 = 7\sin \theta $ with $\frac{\pi }{2}{\text{ < }}\theta {\text{ < }}\pi $
for $\sin 2\theta $ i got $\frac{{( - 2x + 6)\sqrt { - {x^2} - 6x +... | HINT: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
$\cos^2(\theta) + \sin^2(\theta) = 1$
$\frac{\pi}{2} < \theta < \pi$
$\cos(\theta) < 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/865415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$? I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of integration by parts. However, I don't think that's th... | Notice $$\frac{x^2+2x}{\color{blue}{x^4+x^2+1}}
= \frac{x(x+2)}{\color{blue}{(x^2+1)^2-x^2}} =
\frac12\left(\frac{x+2}{x^2-x+1} - \frac{x+2}{x^2+x+1}\right)\\
= \frac12\left(\frac{(x-\frac12)+\frac52}{(x-\frac12)^2+\frac34}
- \frac{(x+\frac12)+\frac32}{(x+\frac12)^2+\frac34}
\right)
$$
Plug this into original integral ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Line not intersecting circle, maximum value of expression involving radius If line $y+x=2$ do not intersect any member of circles $x^2 + y^2 -ax = 0$ at two distinct points where a is parameter, then maximum value of $|a + 4|$.
My try:
Since the line does not intersect the circle at 2 distinct points therefore the dis... | Use substitution to solve the system of equations:
\begin{align*}
\begin{cases}
x^2 + y^2 - ax = 0 \\
y + x = 2
\end{cases}
&\implies x^2 + (2 - x)^2 - ax = 0 \\
&\implies x^2 + (4 - 4x + x^2) - ax = 0 \\
&\implies 2x^2 + (-a - 4)x + 4 = 0 \\
\end{align*}
But since there is at most one intersection point, we know that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve a system of three nonlinear equation in a simple way Given the system:
$$
\begin{cases}
x^2y^2+x^2z^2=axyz & \\
y^2z^2+y^2x^2=bxyz &\\
z^2x^2+z^2y^2=cxyz
\end{cases}
$$
The solution could be gotten in a very tedious way. Is it possible to solve it considering some symmetry property of t... | If is clear we cannot have some $a,b,c$ positive while the rest negative. Otherwise,
$xyz = 0$ and we only have trivial solution.
For simplicity, I will assume $a, b, c > 0$.
If any one of $x,y,z$ vanishes, say $x = 0$, it is clear the equations implies $y z = 0$. This means if one of $x,y,z$ vanish, at least two of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Square of $7+\sum_{k=1}^n6\times10^k$ If we build a number as follow:
$$N=7+\sum_{k=1}^n6\times10^k$$
we find:
$$N^2=9+\sum_{k=1}^n8\times10^k+\sum_{j=n+1}^{2n+1
}4\times10^{j}$$
that means for example:
$67^2=4489$,
$667^2=444889$,
$6667^2=44448889$
and so on. How can we prove that given an arbitrary $n$ and a number... | We have
$$N=1+6\sum_{k=0}^n10^k=1+6\cdot \frac{10^{n+1}-1}9=\frac{2\cdot 10^{n+1}-1}3.$$
Hence
$$N^2=\frac{4\cdot 10^{2n+2}-4\cdot 2^{n+1}+1}{9}=4\cdot 10^{n+1}\cdot \frac{10^{n+1}-1}{9} +8\cdot \frac{10^{n+1}-1}{9}+1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/873442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Where find point $M$ on diagonal of cuboid $ABCDPQRS$ such that the sum $MA+MD$ is minimal? Let a rectangular cuboid of bases $ABCD$ i $PQRS$ such that $AB=1 , BC=2 , AP=3$ . How find on the diagonal $BS$ of the cuboid a point $M$ such that the sum $MA+MD$ is minimal?
| Let $A = (0,0,0)$, $B = (1,0,0)$, $C = (1,2,0)$, $R = (1,2,3)$, $D = (0,2,0)$, $P = (0,0,3)$, $Q = (1,0,3)$, $S = (0,2,3)$. Point $M$ can be expressed as: $t(1,0,0) + (1-t)(0,2,3) = (t,2-2t,3-3t)$, $t \in [0,1]$. Thus: $MA + MD = \sqrt{(t-0)^2 +(2-2t-0)^2 + (3-3t-0)^2} + \sqrt{(t-0)^2 +(2-2t-2)^2 + (3-3t-0)^2} = \sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/874490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $ Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$
I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a multiple of $6$.
| This technique is known as the Roots of Unity Filter. See this related question.
Note that $(1+x)^{n} = \displaystyle\sum_{k = 0}^{n}\dbinom{n}{k}x^k$. Let $\omega = e^{i2\pi/3}$. Then, we have:
$(1+1)^{n} = \displaystyle\sum_{k = 0}^{n}\dbinom{n}{k}1^k$
$(1+\omega)^{n} = \displaystyle\sum_{k = 0}^{n}\dbinom{n}{k}\omeg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/875153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 1
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Find optimal least square solution to the normal equation What is the optimal solution for $\beta_1$ and $\beta_2$ in the following normal equation:
$$\beta _{ 1 }\sum _{ i=1 }^{ n }{ { x }_{ i } } +\beta _{ 0 }=\sum _{ i=1 }^{ n }{ { y }_{ i } } $$
EDIT
Suppose you are given a set of data $(x_i,y_i)$ with y = $\beta_1... | Given a sequence of measurements, $\left\{ x_{k}, y_{k} \right\}_{k=1}^{m}$, and a trial function
$$
y(x) = \beta_{0} + \beta_{1} x,
$$
the linear system is
$$
\begin{align}
\mathbf{A} \beta &= y \\
\left[
\begin{array}{cc}
1 & x_{1} \\
1 & x_{2} \\
\vdots & \vdots \\
1 & x_{m}
\end{array}
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/875376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $a^2=b^2+c^2$ and $0If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove
(a) if $n>2$ then $a^n>b^n+c^n$,
(b) if $0<n<2$ then $a^n<b^n+c^n$.
Part (a) was easy to prove: $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, so $a>b$ and $a>c$. Then O can show: $b^n+c^n=b^2(b^{n-2})+c^2(c^{n-2})<b^2(a^{n-2})+c... | Like you did in (a), we can do as fellows: when $0<n<2$
$$a^n=\frac{a^2}{a^{2-n}}=\frac{b^2}{a^{2-n}}+\frac{c^2}{a^{2-n}}<\frac{b^2}{b^{2-n}}+\frac{c^2}{c^{2-n}}=b^n+c^n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/875549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find the value of this infinitely nested radical (it appears to obtain multiple values)
Find the value of $$\sqrt{1-\sqrt{\frac{17}{16}-\sqrt{1-\sqrt{\frac{17}{16}-\cdots}}}}$$
This is not as simple as it looks for one reason - there are $2$ real solutions to the equation $$x=\sqrt{1-\sqrt{\frac{17}{16}-x}}\implies\b... | Solution technique:
Let
$$L = \sqrt{1 - \sqrt{\frac{17}{6} - \sqrt{1 - \sqrt{\frac{17}{6} -\sqrt{1 - \sqrt{\frac{17}{16} -... } } } }} } $$
The natural method of solution is to observe that
$$L^2 = 1 - \sqrt{\frac{17}{6} - \sqrt{1 - \sqrt{\frac{17}{6} -\sqrt{1 - \sqrt{\frac{17}{16} -... } } } }}$$
So...
$$ L^2 - 1= -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Find $\tan x $ if $\sin x+\cos x=\frac12$ It is given that $0 < x < 180^\circ$ and $\sin x+\cos x=\frac12$, Find $\tan x $.
I tried all identities I know but I have no idea how to proceed. Any help would be appreciated.
| Here is a geometric way of looking at this problem. I will change the angle variable to $t.$ We need to find $\tan t$ if $\sin t + \cos t = 1/2.$ I will use $x = \cos t, y = \sin t$ so we need to solve $$x^2 + y^2 = 1, x + y = 1/2$$ for $x, y.$ that is, to find the two points where the line cuts the unit circle. By el... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 6
} |
Find all $n$ for which $2^n \ge (n+1)^2$
Find all of the elements of $X= \{ n \in \mathbb N: 2^n \ge (n+1)^2\}$
Could someone give me a hint to nudge me in the right direction?
| $\color{green}{2^0=1\ge(0+1)^2=1}$ ?*
$\color{red}{2^1=2\ge(1+1)^2=4}$ ?
$\color{red}{2^2=4\ge(2+1)^2=9}$ ?
$\color{red}{2^3=8\ge(3+1)^2=16}$ ?
$\color{red}{2^4=16\ge(4+1)^2=25}$ ?
$\color{red}{2^5=32\ge(5+1)^2=36}$ ?
$\color{green}{2^6=64\ge(6+1)^2=49}$ ?
$\color{green}{2^7=128\ge(7+1)^2=64}$ ?
$\color{green}{2^8=256\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/878731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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How do you use the BBP Formula to calculate the nth digit of π? I know what the Bailey-Borweim-Plouffe Formula (BBP Formula) is—it's $\pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right]$— but how exactly do I use it to calcul... | The basic idea depends on the following easy result:
The $d+n$-th digit of a real number $\alpha$ is obtained by computing the $n$-th digit of the fractional part of $b^d \alpha$, in base $b$ . (fractional part denoted by $\lbrace \rbrace$.)
For instance:
if you want to find the $13$-th digit of $\pi$ in base $2$, you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
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Finding a limit to negative infinity with square roots: $\lim\limits_{x\to -\infty}(x+\sqrt{x^2+2x})$ Find the limit of the equation
$$\lim_{x\to-\infty} (x+\sqrt{x^2 + 2x})$$
I start by multiplying with the conjugate:
$$\lim_{x\to-\infty} \left[(x+\sqrt{x^2 + 2x})\left({x - \sqrt{x^2 + 2x}\over x - \sqrt{x^2+2x}}\righ... | Hint :
Set $x=-y$, then we will have
$$
\lim_{y\to\infty} \left(\sqrt{y^2 - 2y}-y\right)
$$
Now, repeat the process as you have done (multiplying by its conjugate and dividing by the highest power of denominator).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/881145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Find the smallest positive number $p$ for which the equation $\cos(p\sin x)=\sin(p \cos x)$ has a solution $x\in[0,2\pi].$ Find the smallest positive number $p$ for which the equation
$\cos(p\sin{x})=\sin(p\cos{x})$ has a solution $x$ belonging $[0,2\pi]$.
I am not able to solve this problem.
Please help me.
| It must be that $p\sin{x}+p\cos{x}=\dfrac{\pi}{2}$ with, $p=\dfrac{\pi}{2(\sin{x}+\cos{x})}$.
So, to minimize $p$, $\sin{x}+\cos{x}$ must be maximized.
$\sin{x}+\cos{x}=\sqrt{2} \sin\left(x+\dfrac{\pi}{4}\right)$, which is maximized when $\sin\left(x+\dfrac{\pi}{4}\right)=1$ at $x=\dfrac{\pi}{4}, \dfrac{7\pi}{4}$.
Hen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/881892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Decompose a fraction in a sum of two Let's say that I have this fraction:
$$ \frac{2x}{x^2+4x+3}$$
I would like to decompose in two fraction:
$$ \frac{A}{x+3} + \frac{B}{x+1}$$
Which is the procedure for that? :)
| Hint:
$$\frac{A}{x+3} + \frac{B}{x+1} = \frac{A(x+1) + B(x+3)}{(x+1)(x+3)} = \frac{(A+B)x + (A+3B)}{x^2+4x+3} = \frac{2x}{x^2+4x+3}$$
What must the values of $A$ and $B$ so that $A+B = 2$ and $A + 3B = 0$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/882733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Evaluate the limit of $\ln(\cos 2x)/\ln (\cos 3x)$ as $x\to 0$ Evaluate Limits
$$\lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$$
Method 1 :Using L'Hopital's Rule to Evaluate Limits (indicated by $\stackrel{LHR}{=}$. LHR stands for L'Hôpital Rule)
\begin{align*}
\lim _{x\to \:0}\left(\frac{\ln \left(\cos \left(2x\... | Once you apply L'Hopital's Rule once, and simplify you get $\displaystyle \lim_{x \to 0}\frac{-\frac{2\sin \left(2x\right)}{\cos \left(2x\right)}}{-\frac{3\sin \left(3x\right)}{\cos \left(3x\right)}} = \lim_{x \to 0}\dfrac{2\tan 2x}{3 \tan 3x}$.
Now, applying L'Hopital's Rule one more time gives you $\displaystyle\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/883053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Show that the inequality holds $\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{7}{12}$ We have to show that:
$\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{7}{12}$
To be honest I don't have idea how to deal with it. I only suspect there will be need to consider two cases for $n=2k$ and $n=2k+1$
| Let $k$ be a positive integer and let $x$ be such that $x \in [k-1,k]$. You have
$$
n+k-1 \leq x +n
$$ and
$$
\int_{k-1}^k \frac{1}{x+n} dx \leq \int_{k-1}^k \frac{1}{n+k-1} dx =\frac{1}{n+k-1}
$$ thus, summing from $k=1$ to $n+1$, $n\geq1$, we get
$$
\int_{0}^{n+1} \frac{1}{x+n} dx \leq \sum_{k=1}^{n+1}\frac{1}{n+k-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/883670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 5
} |
Asymptotic behavior of $\sum\limits_{k=1}^n \frac{2^k}{k}$ I'm looking for an asymptotic equivalent of
$$\sum_{0 < k \le n} \frac{2^k}{k}$$
as $n \to \infty$. A plausible candidate seems to be $\frac{2^{n+1}}{n+1}$ (WolframAlpha plot, and the intuitive similarity with $\sum_{k \le n} 2^k = 2^{n+1}$ is also appealing), ... | The Euler-Maclaurin summation formula is useful for approximating sums and often reveals the asymptotic behavior with only a few terms. This problem is an interesting application because the precise asymptotic behavior requires summing an infinite number of terms with Bernoulli numbers as coefficients - the terms tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 0
} |
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: $$\frac{1+x+iy}{1-x-iy}=\... | This reduces to:
$$1+8x+28x^2+56x^3+70x^4+56x^5+28x^6+8x^7+x^8=1-8x+28x^2-56x^3+70x^4-56x^5+28x^6-8x^7+x^8$$Which becomes $$16x+112x^3+112x^5+16x^7=0$$
Whence $x=0$ or $$x^6+7x^4+7x^2+1=0$$
Write $y=x^2$ to obtain $$y^3+7y^2+7y+1=0$$
We have the obvious solution $y=-1$ and $$y^2+6y+1=0$$ for the others
| {
"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": 3
} |
Solve the equation $(x^2-9)+\sqrt{2-x}=0$ Solve the equation $(x^2-9)+\sqrt{2-x}=0$
*
*$(x+3)(x-3)+\sqrt{2-x}=0$
Conditions: $x\neq\pm3 \wedge x\leq2$
*$(x+3)(x-3) = -\sqrt{2-x}$
*$(x+3)^2(x-3)^2 = 2-x$
*$x^4-18x^2+81 = 2-x$
*$x^4-18x^2+x+79 = 0$
I'm positive this isn't going to the right direction.
Please... | Consider making a sign diagram. You will get your answer if you fill in (x+3)(x-3) and ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/892816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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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.