Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$ Inadvertently, I find this interesting inequality. But this problem have nice solution?
prove that
$$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$
This problem have nice solution? Thank you.
ago,I find this
$$\ln{2}<\left(\dfrac{1}{2}\right)^{\frac{1}{2}}=\dfrac{\sqrt{2}}{2... | Using the Taylor series
$$\ln \frac{1+x}{1-x} = \ln (1 + x) - \ln(1 - x) = 2\sum_{k=0}^\infty \frac{x^{2k+1}}{2k+1}, \quad -1 < x < 1;$$
letting $x = 1/3$, we have
$$\ln 2 = 2\sum_{k=0}^\infty \frac{1}{3^{2k+1}(2k+1)}. \tag{1}$$
Using Bernoulli inequality, we have, for all $a > 0$,
$$(2/5)^{2/5} = \frac{1}{a^2}\left(\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "95",
"answer_count": 8,
"answer_id": 6
} |
Integral of $\sin x \cdot \cos x$ I've found 3 different solutions of this integral. Where did I make mistakes? In case there is no errors, could you explain why the results are different?
$ \int \sin x \cos x dx $
1) via subsitution $ u = \sin x $
$ u = \sin x; du = \cos x dx \Rightarrow \int udu = \frac12 u^2 \Righta... | A primitive is unique up to a constant
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/381243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 3
} |
Use the binomial theorem to expand How can we expand this using the binomial theorem?
$(x^2 + 1/x)^7$
| Let $a = x^{2}$ and $b = \frac{1}{x}$. The binomial is written as $(a + b)^{7}$. Apply Binomial Theorem, so we have:
$$(a + b)^{7} = \dbinom{7}{0} a^{7}b^{7 - 7} + \dbinom{7}{1} a^{6}b^{7 - 6} + \dbinom{7}{2} a^{5}b^{7 - 5} + \dbinom{7}{3} a^{4}b^{7 - 4} + \dbinom{7}{4} a^{3}b^{7 - 3} + \dbinom{7}{5}a^{2}b^{7 - 2} + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/383471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving a trig infinite sum using integration How can I prove the following using integration and elementary functions?
Prove that:
$$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$
$0 < \theta < 2\pi$
| Let,
$$S_1 = \sum_{n=1}^{\infty}\frac{\cos n\theta}{n}\\
S_2 = \sum_{n=1}^{\infty}\frac{\sin n\theta}{n}$$
Then $$S_1 + iS_2 = \sum_{n=1}^{\infty}\frac{\cos(n\theta)+i\sin(n\theta)}{n}=\sum_{n=1}^{\infty}\frac{e^{in\theta}}{n}$$
Now, from the Taylor expansion, $\ln (1+x) = x -\frac{x^2}{2}+\frac{x^3}{3} ...$
$$\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
How do you calculate $25^{11} \pmod{341}$? How do you calculate $25^{11} \pmod{341}$?
I understand you have to split the exponent into $11 = 1 + 2 + 8$?
| Note that $ 341 = 11 \cdot 31 $ and hence it is sufficient to fine the residues of $ 25^{11} $ modulo $ 11 $ and $ 31 $ and then apply the Chinese Remainder Theorem.
By Fermat's Little Theorem, $ 25^{10} \equiv 1 \mod 11 \implies 25^{11} \equiv 25 \equiv 3 \mod 11 $.
Also note that $ 25^4 \equiv (-6)^4 \equiv 5^2 \equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/387098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Let M, K, and L be points on line (AB), (BC), and (CA), respectively. Find the maximum area of smallest the three triangles MAL, KBM, and LCK? Let M, K, and L be points on line (AB), (BC), and (CA), respectively. Find the maximum area of smallest the three triangles MAL, KBM, and LCK in respect to ABC?
I try to gues... | We shall denote the area of a triangle $ABC$ with $S_{ABC}$. We shall prove that $\min(S_{MAL}, S_{KBM}, S_{LCK}) \leq \frac{1}{4}S_{ABC}$.
Without loss of generality assume that $S_{MAL}=\max(S_{MAL}, S_{KBM}, S_{LCK})$. Clearly if $\frac{S_{MAL}}{S_{ABC}} \leq \frac{1}{4}$, we are done. Otherwise consider $\frac{S_{M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$ Please help me to find a closed form for the infinite product
$$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$
where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
| Let
$$
f(x)=\prod_{n=0}^\infty\left(1-x^{2^n}\right)^{1/2^n}\tag{1}
$$
and
$$
g(x)=\prod_{n=0}^\infty\left(1+x^{2^n}\right)^{1/2^n}\tag{2}
$$
Then
$$
\begin{align}
f(x)\,g(x)
&=\prod_{n=0}^\infty\left(1-x^{2^{n+1}}\right)^{1/2^n}\\
&=\prod_{n=1}^\infty\left(1-x^{2^n}\right)^{2/2^n}\\
&=\left(\frac{f(x)}{1-x}\right)^2\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 2,
"answer_id": 1
} |
Integral of $\int \frac{x^4+2x+4}{x^4-1}dx$ I am trying to solve this integral and I need your suggestions.
$$\int \frac{x^4+2x+4}{x^4-1}dx$$
Thanks
| Welcome to Math Stack Exchange!
With lab bhattacharjee's helpful hint for the integral, we can rewrite the expression as:
$$\int \left(1 + \frac{(-2x - 5)}{2(x^2 + 1)} + \frac{7}{4(x - 1)} - \frac{3}{4(x + 1)}\right)dx$$
$$= \int 1 dx + \int \frac{(-2x - 5)}{2(x^2 + 1)} dx + \int \frac{7}{4(x - 1)} dx - \int \frac{3}{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Integrate by parts: $\int \ln (2x + 1) \, dx$ $$\eqalign{
& \int \ln (2x + 1) \, dx \cr
& u = \ln (2x + 1) \cr
& v = x \cr
& {du \over dx} = {2 \over 2x + 1} \cr
& {dv \over dx} = 1 \cr
& \int \ln (2x + 1) \, dx = x\ln (2x + 1) - \int {2x \over 2x + 1} \cr
& = x\ln (2x + 1) - \int 1 - {1 \over... | $$ = x\ln (2x + 1) + \ln |{(2x + 1)^{{1 \over 2}}}| - x + C $$
$$ = \ln(2x + 1)^x + \ln(2x + 1)^\dfrac 12 - x + C $$
$$= \ln({(2x + 1)^x \cdot(2x + 1)^\dfrac 12}) - x + C $$
$$= \ln{(2x + 1)^\dfrac{2x+1}{2} } - x + C $$
$$= \dfrac {1}{2}\cdot (2x+1) \ln{(2x + 1)} - x + C $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/393929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
If $\omega$ is a complex cube root of unity, show that the following equals null matrix.
If $\omega$ is a complex cube root of unity, show that
$$ \left( \begin{bmatrix}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & \omega \\
\end{bmatrix} + \begin{bmatrix}
... | $$\omega^3=1 \Rightarrow ω^3−1=0 \Rightarrow (ω−1)(ω^2+ω+1)=0 \Rightarrow ω−1=0 \lor ω^2+ω+1= 0$$
Since, $\omega \neq 1 $ as it is complex , $ω^2+ω+1=0$
Hence, LHS = RHS
- Answer originally posted as comment by Git Gud
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/396248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Explain the 1 + 2 + 3 in $ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{1}{(n+1)/2} $ $$ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{1}{(n+1)/2} = 0 $$
If $n$ goes to infinity, we can image that a bit by taking a very big number.
Like $1.000.000.000$
$1.000.00... | They're using that $$1+2+3+\dots+n=\frac{n(n+1)}2$$
For example $1+2+3=6=\dfrac{3\cdot 4}{2}$
On the other hand $$\underbrace{1+1+1+\cdots+1}_{n \;\;\rm times}=n$$
Thus $$\frac{1+1+1+\cdots}{1+2+3+\cdots}=\frac{n}{\dfrac{n(n+1)}2}=\frac{2}{n+1}$$
Of course, we're interpreting $$\frac{1+1+1+\cdots}{1+2+3+\cdots}$$ as $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/396810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
$ e^{At}$ for $A = B^{-1} \lvert \cdots \rvert B $ For a homework problem, I have to compute $ e^{At}$ for
$$ A = B^{-1} \begin{pmatrix}
-1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3 \end{pmatrix} B$$
I know how to compute the result for $2 \times 2$ matrices where I can calculate the eigenvalues, but this is $3 \times 3$, and... | Joseph is right and $ D^n=\begin{pmatrix}(-1)^n&0&0\\0&2^n&0\\0&0&3^n \end{pmatrix}so (Dt)^n=\begin{pmatrix}(-t)^n&0&0\\0&(2t)^n &0\\0&0&(3t)^n\end{pmatrix}$ so $\left(I+Dt+\frac{1}{2!}(Dt)^2+\frac{1}{3!}(Dt)^3+\cdots\right)=\begin{pmatrix}e^{-t}&0&0\\0&e^{2t}&0\\0&0&e^{3t}\end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/397393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Let $v_1 = (1, 0); v_2 = (1,-1) \space\text{and} \space v_3 = (0, 1).$ I am stuck on the following problem :
Let $v_1 = (1, 0); v_2 = (1,-1) \space\text{and} \space v_3 = (0, 1).$ How many linear transformations
$T \colon \Bbb R^2 \to \Bbb R^2$ are there such that $Tv_1 = v_2; Tv_2 = v_3$ and $Tv_3 = v_1?$ The opt... | Here's a straight forward solution. Every linear transformation $T\colon \mathbb R^2 \longrightarrow \mathbb R^2$ can be represented as a $2\times 2$ matrix, so you want to find $a_1$, $a_2$, $a_3$ and $a_4$ such that
\[\begin{pmatrix}a_1 & a_2 \\ a_3 & a_4\end{pmatrix} \begin{pmatrix}1 \\ 0\end{pmatrix} = \begin{pmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/398018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that a map is Injective How can I prove that $f: (0, \infty) \times (0,\pi) \to \mathbb{R}^2$ where $f(x,y) = (\sinh(x)\sin(y),\cosh(x)\cos(y))$ is injective?
| Suppose $\cosh a\cos b=\cosh c\cos d$ and $\sinh a\sin b=\sinh c\sin d$. Square the two relations:
\begin{gather}
\cosh^2 a\cos^2 b=\cosh^2 c\cos^2 d\\
\sinh^2 a\sin^2 b=\sinh^2 c\sin^2 d
\end{gather}
Sum and subtract the two:
\begin{gather}
\cosh^2 a + \sinh^2 a = \cosh^2 c + \sinh^2 c\\
\cos^2 b - \sin^2 b = \cos^2 d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$ Where do I start to solve a equation for x like the one below?
$$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$
After squaring it, it's too complicated; but there's nothing to factor or to expand?
Ideas?
| $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$
$$\implies \sqrt{7x-4}+\sqrt{4x-2}=\sqrt{4x-1}+\sqrt{7x-5} $$
Squaring we get, $$7x-4+4x-2+2\sqrt{(7x-4)(4x-2)}=4x-1+7x-5+2\sqrt{(4x-1)(7x-5)}$$
$$\implies \sqrt{(7x-4)(4x-2)}= \sqrt{(4x-1)(7x-5)}$$
$$\text{Squaring we get, } (7x-4)(4x-2)=(4x-1)(7x-5)$$
$$\text{ On s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
} |
Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$ Find the limit without the use of L'Hôpital's rule or Taylor series
$$\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$$
| We will use that $\lim\limits_{x\to0}\frac{\sin(x)}{x}=1$, which can be shown geometrically.
First note that
$$
\begin{align}
\frac1{\vphantom{()^2}x^2}-\frac1{\sin^2(x)}
&=\frac{\sin^2(x)-x^2}{x^2\sin^2(x)}\\
&=\frac{\sin^2(x)-x^2}{x^4}\left(\frac{\sin(x)}{x}\right)^{-2}\\
&=\frac{\sin(x)-x}{x^3}\left(\frac{\sin(x)}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
If $17 \mid \frac{n^m - 1}{n-1}$ find the values of $n$ where $m$ is even but not divisible by $4$
Let $m, n \in \mathbb{Z}_+$ with $n > 2$, and let $\frac{n^m-1}{n-1}$ be divisible by $17$. Show that either $m$ is even:$ m \equiv 0 \mod 17$ and $n \equiv 1 \mod 17$. Find all possible values of $n$ in the cases when $... | As the first case is perfect, let me start with the second.
As $17$ is prime and hence $\phi(17)=16$
As Primitive Root is a Generator of Reduced Residue System, let's find a primitive root of $17$
$2^4\equiv-1, 2^8\equiv(-1)^2\equiv1\pmod{17} \implies \text{ord}_{17}2=8<\phi(17)$
$3^2\equiv9, 3^4\equiv-4,3^8=(3^4)^2\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/401462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Telescoping sum of powers
$$
\begin{array}{rclll}
n^3-(n-1)^3 &= &3n^2 &-3n &+1\\
(n-1)^3-(n-2)^3 &= &3(n-1)^2 &-3(n-1) &+1\\
(n-2)^3-(n-3)^3 &= &3(n-2)^2 &-3(n-2) &+1\\
\vdots &=& &\vdots & \\
3^3-2^3 &= &3(3^2) &-3(3) &+1\\
2^3-1^3 &= &3(2^2) &-3(2) &+1\\
1^3-0^3 &= &3(1^2) &-3(1) &+1\\
\underline{\hphantom{(... | $(x-1)^3=x^3-3x^2+3x-1$; now set $x=n-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/402445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Finding the fraction $\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}$ when knowing the sums $a+b+c+d$ to $a^4+b^4+c^4+d^4$ How can I solve this question with out find a,b,c,d
$$a+b+c+d=2$$
$$a^2+b^2+c^2+d^2=30$$
$$a^3+b^3+c^3+d^3=44$$
$$a^4+b^4+c^4+d^4=354$$
so :$$\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}=?$$
If the qusetion i... | If you inspect the system you can see that the solutions for a,b,c,d are all going to have to be small integers. This constrains the system heavily.
I managed to solve the system guessing some small numbers, testing them and tweaking minus signs to get the desired result.
After we have found a,b,c,d the fraction becom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 3
} |
find the value of $\int \frac {x^2}{({x\sin x+\cos x})^2}\,dx$ In my textbook this question solved in this way:
they take since $$\dfrac{d}{dx}(x\sin x+\cos x)=x\cos x $$ so,
$$\int \dfrac {x^2}{({x\sin x+\cos x})^2}\,dx$$
$$\int \dfrac {x\cos x}{({x\sin x+\cos x})^2}\cdot \dfrac {x}{\cos x}\,dx$$
Is there any other w... | Let $u=x\sin{x}+\cos{x}$ so that $du=x\cos{x}~dx$. Note that we chose this since $u$ is composed within another function. Unfortunately, $x\cos{x}$ doesn't quite appear in the integral, but's let's use wishful thinking and pretend that it did. Thus, we can easily solve an integral of the following form:
$$
\int \dfrac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A tricky logarithms problem? $ \log_{4n} 40 \sqrt{3} \ = \ \log_{3n} 45$. Find $n^3$.
Any hints? Thanks!
| Here's a kind of ugly way of doing it:
$$\begin{align*}
(12n)^{\log_{4n}(40\sqrt{3})}&=(12n)^{\log_{3n}(45)}\\\\
(3)^{\log_{4n}(40\sqrt{3})}(4n)^{\log_{4n}(40\sqrt{3})}&=(4)^{\log_{3n}(45)}(3n)^{\log_{3n}(45)}\\\\
(3)^{\log_{4n}(40\sqrt{3})}\cdot 40\sqrt{3}&=(4)^{\log_{3n}(45)}\cdot 45\\\\
(3)^{\log_{4n}(40\sqrt{3})+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/404389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$? I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$,
where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$.
Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ where $s_k$ satisfies the recurrence relation
\begin{align}
& s_{... | Write down the function
$$ g(z) = \sum_{n\geq1} \frac{z^n}{n}H_n^2, $$
so that $S=g(-1)$ and $g$ can be reduced to
$$ zg'(z) = \sum_{n\geq1} z^n H_n^2 = h(z). $$
Now, using $H_n = H_{n-1} + \frac1n$ ($n\geq2$), we can get a closed form for $h(z)$:
$$h(z) = z + \sum_{n\geq2}\frac{z^n}{n^2} + \sum_{n\geq 2}z^n H_{n-1}^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/405356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 5,
"answer_id": 1
} |
What is $\lim\limits_{n→∞}(\frac{n-x}{n+x})^{n^2}$? What is $$\lim_{n\rightarrow\infty}\left(\frac{n-x}{n+x}\right)^{n^2},$$ where $x$ is a real number. Mathematica tells me the limit is $0$ when I put an exact value for $x$ in (Mathematica is inconclusive if I don't substitute for $x$), but using $f(n)=(n-x)^{n^2}$ an... | The interesting form is
$\left(\frac{n-x}{n+x}\right)^{n}$.
The $n^2$ just blows things up.
Taking the log,
$n \ln \frac{n-x}{n+x}
= n \ln \frac{1-x/n}{1+x/n}
= n \big(\ln (1-x/n)- \ln(1+x/n)\big)
$.
Using $\ln(1+z) = z-z^2/2+z^3/3-z^4/4+...$
and
$\ln(1-z) = -z-z^2/2-z^3/3-z^4/4+...$,
and writing $z$ for $x/n$,
this is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
Evaluating $\int_{-1}^{1}\frac{\arctan{x}}{1+x}\ln{\left(\frac{1+x^2}{2}\right)}dx$ This is a nice problem. I am trying to use nice methods to solve this integral, But I failed.
$$\int_{-1}^{1}\dfrac{\arctan{x}}{1+x}\ln{\left(\dfrac{1+x^2}{2}\right)}dx, $$
where $\arctan{x}=\tan^{-1}{x}$
mark: this integral is my favo... | Here is a solution that only uses complex analysis:
Let $\epsilon$ > 0 and consider the truncated integral
$$ I_{\epsilon} = \int_{-1+\epsilon}^{1} \frac{\arctan x}{x+1} \log\left( \frac{1+x^2}{2} \right) \, dx. $$
By using the formula
$$ \arctan x = \frac{1}{2i} \log \left( \frac{1 + ix}{1 - ix} \right) = \frac{1}{2i}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 2
} |
Product of nilpotent matrices. Let $A$ and $B$ be $n \times n$ complex matrices and
let $[A,B] = AB - BA$.
Question: If $A , B$ and $[A,B]$ are all nilpotent matrices,
is it necessarily true that $\operatorname{trace}(AB) = 0$?
If,in fact, $[A,B] = 0$, then we can take $A$ and $B$ to be strictly upper triangular ... | No. While the conjecture is true for $n=2$ and computer experiments suggest that it may also be true for $n=3$, counterexamples are abundant when $n=4$. Let $A$ be the $4\times4$ Jordan block. The following $B$s are some computer generated random counterexamples that satisfy the condition $B^4=(AB-BA)^4=0$ but $\mathrm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/408499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$ Can you help me to prove that
$$(x+y)^n\leq 2^{n-1}(x^n+y^n)$$
for $n\ge1$ and $x,y\ge0$.
I tried by induction, but I didn't get a result.
| Look. I also have tried to do it by induction. It is obvious that it holds for $n=1$ and $n=2$. Assume that it also holds for $n$. Let's prove that inequality for $n+1$:
$$
2^n(x^{n+1} + y^{n+1}) - (x+y)^{n+1} = 2^{n}(x^{n+1} + y^{n+1}) -(x+y)^n(x+y)\geq 2^n(x^{n+1} + y^{n+1}) - 2^{n-1}(x^n + y^n)(x+y) =
2^{n-1}(x^{n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 1
} |
Prove by induction that $1^3 + 2^3 + 3^3 + .....+ n^3= \frac{n^2(n+1)^2}{4}$ for all $n\geq1$. Use mathematical induction to prove that $1^3 + 2^3 + 3^3 + .....+ n^3= \frac{n^2(n+1)^2}{4}$ for all $n\geq1$.
Can anyone explain? Because I have no clue where to begin. I mean, I can show that $1^3+ 2^3 +...+ (k+1)^3=\frac{... | Hints:
$$1^3=\frac{1^2\cdot2^2}4\;\;\color{green}\checkmark$$
$$1^3+2^3+\ldots+n^3+(n+1)^3\stackrel{\text{Ind. Hyp.}}=\frac{n^2(n+1)^2}4+(n+1)^3=$$
$$=\frac{(n+1)^2}4\left(n^2+4(n+1)\right)=\ldots\ldots\;\;\;\color{green}\checkmark$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/411485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Limit with roots I have to evaluate the following limit:
$$ \lim_{x\to 1}\dfrac{\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}} . $$
I rationalized both the numerator and the denominator two times, and still got nowhere. Also I tried change of variable and it didn't work.
Any help is gratefu... | Applying L'Hopital's Rule, we get
$$
\lim_{x\rightarrow1}{\frac{\frac{1}{2\sqrt{x+1}}+\frac{2x}{2\sqrt{x^2-1}}+\frac{3x^2}{2\sqrt{x^3+1}}}{\frac{1}{2\sqrt{x-1}}+\frac{2x}{2\sqrt{x^2+1}}+\frac{4x^3}{2\sqrt{x^4+1}}}}.
$$
The $2$ in each denominator cancels, and we multiply the numerator and denominator by
$$
\sqrt{x-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/411676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 5
} |
Find the limit : $\lim\limits_{n\rightarrow\infty}\int_{n}^{n+7}\frac{\sin{x}}{x}\,\mathrm dx$ I have this exercise I don't know how to approach :
Find the limit : $$\lim_{n\rightarrow\infty}\int_{n}^{n+7}\frac{\sin x}{x}\,\mathrm dx$$
I can see that with $n\rightarrow\infty$ the area under the graph of this functio... | No I don't think that there is a lot that can be got out of it. For example the result is not yet sufficient condition for convergence of the integral from $0$ to $\infty$. However, the Riemann improper integral of the first kind
\begin{equation}
\int_0^\infty \textrm{sinc}(x/\pi) dx
\end{equation}
where $\textrm{sinc}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
Limit of $\lim\limits_{n\to \infty}\left(\frac{1^\frac{1}{3}+2^\frac{1}{3}+3^\frac{1}{3}+\dots+n^\frac{1}{3}}{n\cdot n^\frac{1}{3}} \right)$ I want to evaluate this limit.
$$\lim_{n\to \infty}\left(\frac{1^\frac{1}{3}+2^\frac{1}{3}+3^\frac{1}{3}+\dots+n^\frac{1}{3}}{n\cdot n^\frac{1}{3}} \right)$$
What I did is:
set $f... | We have
$$\sum_{k=1}^n k^{\frac{1}{3}}\sim_\infty\int_1^nx^{\frac{1}{3}}dx\sim_\infty\frac{3}{4}n^{\frac{4}{3}}$$
so
$$\lim_{x\to \infty}\left(\frac{1^\frac{1}{3}+2^\frac{1}{3}+3^\frac{1}{3}+\dots+n^\frac{1}{3}}{n\cdot n^\frac{1}{3}} \right)=\frac{3}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/414807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all $1 \leq i \leq n$? I need to prove this result for something I am working on:
For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ non... | Yes.
n is the sum of two squres iff all odd $ 3\mod4$ prime factors of n occur an even number of times and every positive integer is the sum of four possibly zero squares. The only positive integers not expressible as the sum of four non-zero squares are $1,3,5,9,11,17,29,41,2\times 4^n,6\times 4^n, 14\times 4^n$. If ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Given that $x = 4\sin \left( {2y + 6} \right)$ find dy/dx in terms of x My attempt:
$\eqalign{
& x = 4\sin \left( {2y + 6} \right) \cr
& {{dx} \over {dy}} = \left( 2 \right)\left( 4 \right)\cos \left( {2y + 6} \right) \cr
& {{dx} \over {dy}} = 8\cos \left( {2y + 6} \right) \cr
& {{dy} \over {dx}} = {1 \ov... | You got:
$\dfrac{dy}{dx}=\dfrac{1}{8\cos(2y+6)}$
And we have
$x=4\sin(2y+6)\implies y=\dfrac{1}{2}\left(\sin^{-1}\left(\dfrac{x}{4}\right)-6\right)$
Plug that in the top equation:
$\dfrac{dy}{dx}=\dfrac{1}{8\cos\left(\sin^{-1}\left(\dfrac{x}{4}\right)\right)}$
We know $\cos^2x+\sin^2x=1\implies \cos x=\pm\sqrt{1-\sin^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Finding the closest point to the origin of $y=2\sqrt{\ln(x+3) }$ Given $$y=2\sqrt{\ln(x+3) }$$
How do I determine a (x,y) pair satisfying the above relation which is the closest to the origin (0,0)?
| To minimize the distance from the origin, minimize the square of the distance from the origin, given by $d^2 = M = x^2+y^2$,
$$M = x^2+y^2=x^2+(2\sqrt{\ln(x+3)})^2 = x^2 + 4\ln(x+3)$$
$$\frac{dM}{dx} = 2x + \frac{4}{x+3} = \frac{2x(x+3) + 4}{x+3} = \frac{2x^2 + 6x + 4}{x+3}.$$
This is only (possibly) equal to zero when... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Differential equation Show that the differential equation $$\frac{dy}{dx} =\frac{e^x+x}{\sin y+2}$$ has a solution satisfying $y(0) = \pi$. To do this, separate variables and integrate to get an equation implicitly relating $x$ and $y$.
| $$
\frac {dy}{dx} = \frac {e^x + x}{\sin y + 2} \\
(\sin y + 2) dy = (e^x + x) dx \\
\int (\sin y + 2) dy = \int (e^x + x) dx \\
-\cos y + 2y = e^x + \frac {x^2}2 + C
$$
Now, substitute $y(0) = \pi$
$$
1 + 2\pi = 1 + C
$$
from which you can find that $C = 2\pi$.
Final answer is
$$
2y - \cos y = e^x + \frac {x^2}2 + 2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to find the integral of $\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$ $$\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$$
What is the method to find an integral like this?
| $$\int_0^a f(x)\;dx=2\int_0^\dfrac a2 f(x)\;dx\; if \;f(a-x)=f(x)\;$$
and$\int_0^a f(x)\;dx=0\;\;if \;f(a-x)=-f(x)$
here $f(x)=\sin ^6x\implies f(\pi-x)=\sin ^6(\pi-x)\implies \sin^6x\implies f(x)$
so we can write it as first identity:
$$I=\int_0^\dfrac \pi2 \sin^6 x\;dx$$
from identity:$\int_0^af(x)\;dx=\int_0^af{(a-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
Showing that $\int_0^{\pi/3}\frac{1}{1-\sin x}\,\mathrm dx=1+\sqrt{3}$
Show that $$\int_0^{\pi/3}\frac{1}{1-\sin x}\,\mathrm dx=1+\sqrt{3}$$
Using the substitution $t=\tan\frac{1}{2}x$
$\frac{\mathrm dt}{\mathrm dx}=\frac{1}{2}\sec^2\frac{1}{2}x$
$\mathrm dx=2\cos^2\frac{1}{2}x\,\mathrm dt$
$=(2-2\sin^2\frac{1}{2}x)\... | I think you mean
$$\int_0^{\pi/3} \frac{dx}{1-\sin{x}}$$
which may be accomplished using the substitution $t=\tan{\frac{x}{2}}$. Then
$$dt = \frac12 \sec^2{\frac{x}{2}} dx= \frac12 (1+\tan^2{\frac{x}{2}}) dx = \frac12(1+t^2) dx$$
so that $dx = 2 dt/(1+t^2)$ Also
$$t^2=\frac{1-\cos{x}}{1+\cos{x}} \implies \cos{x} = \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Number of solution for $xy +yz + zx = N$ Is there a way to find number of "different" solutions to the equation $xy +yz + zx = N$, given the value of $N$.
Note: $x,y,z$ can have only non-negative values.
| equation:
$XY+XZ+YZ=N$
Solutions in integers can be written by expanding the number of factorization: $N=ab$
And using solutions of Pell's equation: $p^2-(4k^2+1)s^2=1$
$k$ -some integer which choose on our own.
Solutions can be written:
$X=ap^2+2(ak+b+a)ps+(2(a-2b)k+2b+a)s^2$
$Y=2(ak-b)ps+2(2ak^2+(a+2b)k+b)s^2$
$Z=bp^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
$X \sim U [-0.5 , 1.5] , Y = X^2$ Given $$ f(x) = \begin{cases} \frac12 &,\ -0.5 \le x\le 1.5\\0 &,\ \mbox{otherwise} \end{cases}$$
find the probability density function of $Y=X^2$.
To solve this I first divided up the pdf of X into three parts:
$$f(x) = \begin{cases} \frac12 &, \ -0.5 \le x\le 0\\\frac12 &, \ 0\le x\l... | It looks fine. However, for future reference, I highly recommend the following method: find the cdf of $X$, find the cdf of $Y$, differentiate the cdf to get back your pdf for $Y$. The steps for this problem would be as follows:
For $x<.5$, the cdf comes out to zero, and for $x>1.5$, it comes out to $1$. For the val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$
$$\int \dfrac{1}{\cos x+\tan x }dx$$
This can be converted to
$$\int \dfrac{\cos x}{\sin x+\cos^2x}dx$$
But from here I get stuck. Using t substitution will get you into a mess. Are there any tricks which can split the fraction into sim... | As commented above the Weirstrass substitution
$$
\begin{equation*}
t=\tan \frac{x}{2}\Leftrightarrow x=2\arctan t,\,dx=\frac{2}{
1+t^{2}}dt,
\end{equation*}
$$
can be applied to the given integral. The integrand becomes a rational
fraction in $t$, because
$$
\begin{equation*}
\cos x=\frac{1-\tan ^{2}\frac{x }{2}}{1+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Binomial expansion of expression with numerator and denominator both linear equations of x How can we expand the following by the binomial expansion, upto the term including $x^3$? That'll be 4 terms.
This the expression to be expanded: $\sqrt{2+x\over1-x}$
I understand how to do the numerator and denominator indiv... | We will use the expansion
$\sqrt{1+x} = 1+x/2+x^2(1/2)(-1/2)/2 + x^3(1/2)(-1/2)(-3/2)/6 + ...
= 1+x/2-x^2/8+x^3/8+...
$
where "..." means "terms of higher order than $x^3$"
both in this expansion and in the math below.
Note: I am doing the following math
off the top of my head
as I am entering it,
so the chances for er... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
problem with partial fraction decomposition I want to do partial fraction decomposition on the following rational function:
$$\frac{1}{x^2(1+x^2)^3}$$
So I proceed as follows:
$$\begin{align}
\frac{1}{x^2(1+x^2)^3} &= \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{1 + x^2} + \frac{Ex + F}{(1 + x^2)^2} + \frac{Gx + H}{(1 +... | Your final expression is almost correct; you're just forgetting the $F$. The coefficient of $x^4$ should be:
$$
3B + 2D + F
$$
and the coefficient of $x^2$ should be:
$$
3B + D + F + H
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/425021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding $\sin^6 x+\cos^6 x$, what am I doing wrong here? I have $\sin 2x=\frac 23$ , and I'm supposed to express $\sin^6 x+\cos^6 x$ as $\frac ab$ where $a, b$ are co-prime positive integers. This is what I did:
First, notice that $(\sin x +\cos x)^2=\sin^2 x+\cos^2 x+\sin 2x=1+ \frac 23=\frac53$ .
Now, from what was... | Should be $(\sin^2 x+\cos^2 x)^3=1=\sin^6 x+\cos^6 x+3\sin^4 x \cos^2 x+3\cos^4 x \sin^2 x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/425664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
System of three equations in three variables? Fibonacci apparently found some solutions to this problem:
Find rational solutions of:
$$x+y+z+x^2=u^2$$
$$x+y+z+x^2+y^2=v^2$$
$$x+y+z+x^2+y^2+z^2=w^2$$
How would you find solutions to this using the mathematics available in Fibonaccis's time? (of course by this I mostly me... | This is not a full answer in that not all solutions are described. But the discussion yields two infinite parametrized families of solutions. And the methods could possibly be studied longer to find more families, and possibly parametrize all solutions. As proof that this works before you invest in studying it, check t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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How to understand and create quaternions? I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$.
I understand how to do the calculation of quaternions basicly works and how to do the slerp. There is a lot of lite... | Okay, I think I got it. The quaternion is created like this:
$$q = \begin{pmatrix} a_x \cdot \sin{\frac{\alpha}{2}} \\ a_y \cdot \sin{\frac{\alpha}{2}} \\ a_z \cdot \sin{\frac{\alpha}{2}} \\ \cos{\frac{\alpha}{2}} \end{pmatrix}$$
I have one quaternion for each axis. Each axis is a vector (with three coordinates) and is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Factor $x^4 - 11x^2y^2 + y^4$ This is an exercise from Schaum's Outline of Precalculus. It doesn't give a worked solution, just the answer.
The question is:
Factor $x^4 - 11x^2y^2 + y^4$
The answer is:
$(x^2 - 3xy -y^2)(x^2 + 3xy - y^2)$
My question is:
How did the textbook get this?
I tried the following methods (exa... | Let $x^2=u$ and $y^2=v$, then expression becomes $u^2-11uv+v^2$. Let the roots of the quadratic equation $u^2-11uv+v^2=0$ be $\alpha v,\beta v$, then, $\alpha+\beta=11$ and $\alpha\beta=1$ . You can check easily that $\alpha,\beta>0$, so $\sqrt{\alpha},\sqrt{\beta}\in \Bbb R$
therefore, $$x^4-11x^2y^2+y^4=u^2-11uv+v^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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"answer_id": 2
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Limit on the expression containing sides of a triangle To find the bounds of the expression $\frac{(a+b+c)^2}{ab+bc+ca}$, when a ,b, c are the sides of the triangle.
I could disintegrate the given expression as $$\dfrac{a^2+b^2+c^2}{ab+bc+ca} + 2$$ and in case of equilateral triangle, the limit is 3.
Now how to procee... | (Without using the condition that $a, b, c$ are sides of a triangle) We know that
$$(a-b)^2+(b-c)^2+(c-a)^2 \geq 0 $$
This implies that $a^2 + b^2 + c^2 \geq ab + bc + ca$. Hence $\frac{a^2+b^2+c^2} { ab+bc+ca} \geq 1$, so the minimum of the initial expression is 3.
To find the maximum, you will need to use the trian... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
$ 1 + \sqrt{1 - \sqrt{x^4-x^2}} = x$ can be written in the form $\frac{a}{b}$, find $a+b$. The solution to the equation
$$1 + \sqrt{1 - \sqrt{x^4-x^2}} = x$$
can be written in the form $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. Find $a+b$.
| First isolate the square root on one side:
$$\sqrt{1-\sqrt{x^4-x^2}}=x-1\;.$$
Now square:
$$1-\sqrt{x^4-x^2}=(x-1)^2=x^2-2x+1\;.$$
Then $-\sqrt{x^4-x^2}=x^2-2x$, $\sqrt{x^4-x^2}=2x-x^2$, and $x^4-x^2=(2x-x^2)^2=x^4-4x^3+4x^2$. Simplification yields the equation $4x^3-5x^2=0$, or $x^2(4x-5)=0$. This is easy to solve. Be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For real numbers $x$ and $y$, show that $\frac{x^2 + y^2}{4} < e^{x+y-2} $
Show that for $x$, $y$ real numbers, $0<x$ , $0<y$
$$\left(\frac{x^2 + y^2}{4}\right) < e^{x+y-2}. $$
Someone can help me with this please...
| When $x+y$ is fixed $x^2+y^2=(x+y)^2-2xy$ takes maximal value when $xy=0.$ So it is enough to prove our inequality when one variable is $0.$ In this case, it can be reduced to $\frac{x^2}{4}\le e^{x-2}$ or $e^z\ge 1+z+\frac{z^2}{4}$ which immediately follows from the fact that $e^z\ge 1+z+\frac{z^2}{2}.$
Alternative wa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Applonius Circle/ Ford Circle / Infinite GP / Circle Packing
All the smaller circles are mutually tangent and continue to infinity. What is sum of radii of all the smaller circles?
| This is similar to the Pappus chain in the arbelos and can be answered with circle inversion the same way as that problem is analyzed at Cut The Knot.
Inversion of the figure through a circle centered at $A$ through $C$ (dashed red) takes the semicircles on $AB$ and $AC$ to parallel lines perpendicular to $AC$. The im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Probability of randomly generated quadratic equation having equal roots Could any one help me to solve this problem?
Given that the coefficients of the equation $ax^2+bx+c=0$ are selected by throwing an unbiased die, we need to find what is the probability of the equation having equal roots.
Thank you for Hints.
| Hint: The roots are the same iff $b^2-4ac=0$ so find the number of triples $(a,b,c)$ with each of $a,b,c$ in $\{1,2,3,4,5,6\}$ for which this equation holds, then divide by $6^3$.
OP has asked about finding the $a,b,c$: First from $b^2=4ac$ we see that $b$ must be even, so that it is one of $2,4,6$
$b=2$ (so that $b^2=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Covergence of $\frac{1}{4} + \frac{1\cdot 9}{4 \cdot 16} + \frac{1\cdot9\cdot25}{4\cdot16\cdot36} + \dotsb$ I am investigating the convergence of the following series: $$\frac{1}{4} + \frac{1\cdot 9}{4 \cdot 16} + \frac{1\cdot9\cdot25}{4\cdot16\cdot36} + \frac{1\cdot9\cdot25\cdot36}{4\cdot16\cdot36\cdot64} + \dotsb$$
T... | Since
$$
\prod_{k=1}^n\frac{2k-1}{2k}\times\prod_{k=1}^n\frac{2k}{2k+1}=\frac1{2n+1}
$$
and
$$
\begin{align}
1
&\ge\left.\prod_{k=1}^n\frac{2k-1}{2k}\middle/\prod_{k=1}^n\frac{2k}{2k+1}\right.\\
&=\prod_{k=1}^n\frac{4k^2-1}{4k^2}\\
&\ge\prod_{k=1}^\infty\frac{4k^2-1}{4k^2}\\
&=\prod_{k=1}^\infty\left(1+\frac1{4k^2-1}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$ $$\int \frac{dx}{\sqrt{x^2 -9}}$$
$x = 3 \sec \theta \implies dx = 3 \sec\theta \tan\theta d\theta$
$$\begin{align} \int \frac{dx}{\sqrt{x^2 -9}} & = \frac{1}{3}\int \frac{3 \sec\theta \tan\theta d\theta}{\tan\theta} \\ \\ & = \int \sec\theta d\theta \\ \\ & = \ln | \sec\the... | $\theta=arc\sec\frac{x}{3}\implies \sec\theta=\frac{x}{3}$ and then $\tan\theta=\frac{\sqrt{x^2-9}}{3}$ so, integral would be $\ln\left(\frac{x}{3}+\frac{\sqrt{x^2-9}}{3}\right)+c=\ln(x+\sqrt{x^2-9})+k(=-\ln 3+c)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/436153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
$y =f(x) =(ax^2 + bx +c)/(dx^2+ex+f)$ We have to find the conditions for this it takes all real values. $$
y=f(x)=\frac{ax^2+bx+c}{dx^2+ex+f}
$$
We have to find the conditions for this it takes all real values.
MY solution
One approach is to equate it to y and for a quadratic of x and put discriminant greater than equa... | $$f\left( x \right) =\frac { a{ x }^{ 2 }+bx+c }{ d{ x }^{ 2 }+ex+f } =\frac { a{ x }^{ 2 }+bx+c }{ d\left( { x }^{ 2 }+\frac { e }{ d } x+\frac { f }{ d } \right) } =\frac { a{ x }^{ 2 }+bx+c }{ d\left( { x }^{ 2 }+\frac { e }{ d } x+\frac { e^{ 2 } }{ 4{ d }^{ 2 } } -\frac { e^{ 2 } }{ 4{ d }^{ 2 } } +\frac { f }{ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/437450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
How to calculate $\cos(6^\circ)$? Do you know any method to calculate $\cos(6^\circ)$ ?
I tried lots of trigonometric equations, but not found any suitable one for this problem.
| I'm going to use the value of $\cos 18°=\frac{1}{4}\sqrt{10+2\sqrt{5}}$ obtained in this question.
$\sin^2 18°=1-\left(\frac{1}{4}\sqrt{10+2\sqrt{5}}\right)^2=1-\frac{10+2\sqrt{5}}{16}=\frac{6-2\sqrt{5}}{16}$ so $\sin 18°=\frac{1}{4}\sqrt{6-2\sqrt{5}}$
$\sin 36°=2\cos 18°\sin 18°=\frac{1}{4}\sqrt{10-2\sqrt{5}}$
$\cos 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 4
} |
How to find the smallest positive integer $K$ such that $(K -\lfloor\frac{K}{2}\rfloor + 1)(\lfloor\frac{K}{2}\rfloor + 1) \geq N$ I am writing a program and I would need an explicit formula for the following:
The smallest positive integer $K$ such that:
$$\left(K - \left\lfloor\frac{K}{2}\right\rfloor + 1\right)\left(... | Say we have a positive integer $K$ such that
$$\left(K - \left\lfloor\frac{K}{2}\right\rfloor + 1\right)\left(\left\lfloor\frac{K}{2}\right\rfloor + 1\right) \geqslant N.\tag{1}$$
Now, if $K = 2m$ is even, $(1)$ means $$(m+1)^2 \geqslant N \iff m \geqslant \sqrt{N} - 1 \iff K \geqslant 2(\sqrt{N}-1),$$ and since $K$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/439883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Solve the equation $z^3=z+\overline{z}$ I have been trying to solve an equation $z^3=z+\overline{z}$, where $\overline{z}=a-bi$ if $z=a+bi$. But I cant find any clues on how to move forward on that one. Please help.
| This is akin particularly to the methods described by Elias , Sujaan Kunalan and nbubis :
The sum of a complex number and its conjugate produce a pure real number,
$$ z \ + \ \overline{z} \ = \ 2a \ = \ 2a \ \cdot \ cis (0) = \ 2a \ \cdot \ cis (2 \pi) \ , $$
$ \ cis( \theta ) \ $ being the abbreivated form of $ \ \co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/440000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 9,
"answer_id": 6
} |
$a, b, c, d$ are positive integers， $a-c|a b+c d$, and then $a-c|a d+b c$
$a, b, c, d$ are positive integers， $a-c|a b+c d$, and then $a-c|a d+b c$
proof: really easy when use $a b+c d-(a d+b c)$
however my first thought is, $a-c| a b+c d+k(a-c)$, and set some $k$ to prove, failed.
question1 : is this method could b... | As lab gave you the $k$ which works for question 1, here is the answer to question 2:
$$\det \begin{pmatrix} a & d \\-c& b \end{pmatrix} = \det \begin{pmatrix} a & b \\-c & d \end{pmatrix}+\det \begin{pmatrix} a & d-b \\-c & b-d \end{pmatrix}$$
$$ = \det \begin{pmatrix} a & b \\-c & d \end{pmatrix}+(b-d)\det \begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/440496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve these two trigonometric equations? How to solve these two trigonometric equations :
$$\sin y \sin(2x+y)=0$$
$$\sin x \sin(x+2y)=0.$$
I know one set of solution will be $(0,0)$. What will be the other set ?
| A possibility is $\sin y=0$, which gives $y=k\pi$ (integer $k$); substituting in the second one gives
$$
\sin x\sin(x+2k\pi)=0
$$
so this implies $\sin x=0$. Similarly, $\sin x=0$ implies $\sin y=0$. Thus we can reduce to the case
$$\begin{cases}
\sin(2x+y)=0\\
\sin(x+2y)=0
\end{cases}
$$
From this you have
$$
\begin{c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/441133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Are there infinitely many rational outputs for sin(x) and cos(x)? I know this may be a dumb question but I know that it is possible for $\sin(x)$ to take on rational values like $0$, $1$, and $\frac {1}{2}$ and so forth, but can it equal any other rational values? What about $\cos(x)$?
| There are infinitely many primitive Pythagorean triples, that is, triples $(a,b,c)$ of positive integers such that $a$, $b$, and $c$ are positive integers $\gt 1$ such that $a^2+b^2=c^2$.
Any such triple determines a right triangle. The sines and cosines of the two non-right angles are the rationals $\frac{a}{c}$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/442530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 2
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Partial fractions of $\frac{-5x+19}{(x-1/2)(x+1/3)}$ Alright, I need to find the partial fractions for the expression above. I have tried writing this as $$\frac{a}{x-1/2}+\frac{b}{x+1/3}$$ but the results give me $a=25.8$ and $b=-20.8$, which are slightly wrong because they give me $5x+19$ instead of $-5x+19$. Can you... | Either you've started from the wrong equations, or you've made a calculation mistake (don't worry, happens to all of us sometimes). When you write
$$\frac{a}{x-\frac{1}{2}}+\frac{b}{x+\frac{1}{3}}=\frac{-5x+19}{(x-\frac{1}{2})(x+\frac{1}{3})}$$
you get
$$\frac{a(x+\frac{1}{3})}{(x-\frac{1}{2})(x+\frac{1}{3})}+\frac{b(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$
If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$
I have had a few ideas about this:
If $\alpha +\beta = \dfrac{\pi}{4}$ then $\tan(\alpha +\beta) = \tan(\dfrac{\pi}{4}) = 1$
We also know that $\tan... | from where OP left his step:$$1 = \dfrac{\tan\alpha + \tan\beta}{1- \tan\alpha\tan\beta}$$
$$\implies{1- \tan\alpha\tan\beta}=\tan\alpha + \tan\beta$$
$$\implies \tan\alpha + \tan\beta+\tan\alpha\tan\beta=1$$
add 1 to both sides
$$\implies\tan\alpha + \tan\beta+\tan\alpha\tan\beta+1=2$$
$$\implies1+\tan\alpha + \tan\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Stuck at Evaluating the Riemann-Stieltjes Integral Let $$\alpha(x) = \left\lbrace \begin{array}{cc} 0 & x=0\\ \dfrac{1}{2^n} & \dfrac{1}{3^n} < x \leq \dfrac{1}{3^{n-1}}\quad n=1,2,...\end{array}\right.$$
Evaluate $$\int_{0}^{1}{x\mathrm{d}\alpha(x)}$$
Attempt: I dont know how to put this formally, but I know it is sim... | I think your answer is quite right. I'm going to outline an alternative, wimpy approach using integration by parts:
$$\int_0^1 x \, d\alpha(x) = [x \, \alpha(x)]_0^1 - \int_0^1 dx \, \alpha(x)$$
The integrated term is $1/2$, and the resulting integral is straightforward:
$$\int_0^1 x \, d\alpha(x) = \frac12 - \sum_{n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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If $x,y,z \in \Bbb{R}$ such that $x+y+z=4$ and $x^2+y^2+z^2=6$, then show that $x,y,z \in [2/3,2]$. If $x,y,z\in\mathbb{R}$ such that $$x+y+z=4,\quad x^2+y^2+z^2=6;$$then show that the each of $x,y,z$ lie in the closed interval $[2/3,2]$.
I have been able to solve using $2(y^2+z^2)\geq(y+z)^2$.
Is there any another met... | My Solution:: Given $x+y = 4-z$ and $x^2+y^2=6-z^2$.
Now using the Cauchy-Schwarz inequality, we get $(x^2+y^2)\cdot (1^2+1^2)\geq (x+y)^2$
So we get $(12-2z^2)\geq (4-z)^2\Rightarrow 12-2z^2\geq 16+z^2-8z\Rightarrow 3z^2-8z+4\leq 0$
So $\displaystyle (3z^2-6z)-(2z-4)\leq 0\Rightarrow 3z(z-2)-2(z-2)\leq 0\Rightarrow \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Find $F_{n}$ in : $F_{n} +2F_{n-1} + ... + (n+1)\cdot F_{0} = 3^{n}$ I'm stuck with the question for a while : Find in $$F_{n} +2F_{n-1} + ... + (n+1)\cdot F_{0} = 3^{n}$$
the element $F_{n}$ .
Placing $n-1$ instead on $n$ results in :
$$F_{n-1} +2F_{n-2} + ... + (n-1+1)\cdot F_{0} = 3^{n-1}$$
$$ F_{n-1} +2F_{n-2} + .... | Generating functions are your friends.
With experience, you will recognize that sum
as a convolution.
All that follows is standard generating function manipulation.
See the book "generatingfunctionology"
available free online
at
http://www.math.upenn.edu/~wilf/DownldGF.html.
Note: This may seem unnecessarily complicate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/448516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Functional equation: $R(1/x)/x^2 = R(x) $ The following can be shown without much hassle.
Suppose $R$ is a rational function satisfying the following functional equation.
\begin{align}
\frac{1}{x^2} R\left( \frac{1}{x} \right) = R(x) \qquad \forall \: x \in \mathbb{R} \backslash \{0\}
\tag{1}
\end{align}
The... | If
$\frac{1}{x^2} R\left( \frac{1}{x} \right) = R(x)$,
$x^2 R(x) = R(1/x)$.
If $R(x) = A(x)/B(x)$,
where $A$ and $B$ are relatively prime polynomials
of respective degrees $n$ and $m$,
$A(1/x)=a(x)/x^n$ and
$B(1/x) = b(x)/x^m$,
where $a(x)$ and $b(x)$
are the reciprocal polynomials of
$A$ and $B$, respectively.
Then
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 4
} |
Express $f'''_{xxx} and f'''_{yyy}$ in terms of $f'''_{uuu} and f'''_{vvv}$. Let $f(x,y)\in C^3(\mathbf{R}^2)$ and let $u=x+y$ and $v=y$.
Express $f'''_{xxx} and f'''_{yyy}$ in terms of $f'''_{uuu} and f'''_{vvv}$.
I'm supposed to use the chain rule, how do I go about?
Thanks!
Alexander
| If $f$ is of the form $f(x_1,\ldots,x_n)$ I will use the notation $D_if$ for the partial derivativeof $f$ w.r.t. the $i$-th variable. Similarly, $D_{k,i}f=D_kD_if$ denotes the partial derivative of $D_if$ w.r.t. the $k$-th derivative.
Then $$\begin{align}D_1\left(f(x+y,y)\right)&=D_1f(x+y,y)D_1(x+y)+D_2f(x+y,y)D_1(y)\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Improper integral that converges for all $x$ in $ \mathbb{R}$
Let $f(x)$ be defined by the improper integral:
$$f(x)= \int_{0}^{\infty} \cos\left(\frac{t^3}{3} + \frac{x^2 t^2}{2} + xt\right)dt.$$
Show that this improper integral converges for all $x \in\mathbb{R}$.
How do you start this question? Do I evaluate ... | Let $x\in\mathbb R$, then there exists an $N>0$ such that $t^2+x^2t+x>0$ for all $t>N$. Then, if $M>N$: you have that $$\int_N^M\cos\left(\frac{t^3}{3}+\frac{x^2t^2}{2}+xt\right)\,dt=\int_N^M\left(\sin\left(\frac{t^3}{3}+\frac{x^2t^2}{2}+xt\right)\right)'\frac{1}{t^2+x^2t+x}\,dt=\left[\sin\left(\frac{t^3}{3}+\frac{x^2t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Condition for collinearity of points $(a, a^3), (b, b^3), and (c, c^3)$ The following is a statement I have been trying to prove (while solving problem 1.4.26 in Algorithms (4th edition) by Robert Sedgewick).
Show that three points $(a, a^3), (b, b^3), and
(c, c^3)$ are collinear if and only if $a + b + c = 0$.
I ... | Given three distinct points in ${\mathbb{R}^2}$ $P1=(a,a^3)$, $P2=(b,b^3)$ and $P3=(c,c^3)$ for $a,b,c \in \mathbb{R}$, we define ${\vec u},{\vec v} \in \mathbb{R}^2$ such that
$$\vec u = P2-P1=\left( {\begin{array}{*{20}{c}}
{{b^3} - {a^3}} \\
{b - a}
\end{array}} \right), \vec v = P3-P1=\left( {\begin{array}{*{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Simplifying a Rational Expression How do you simplify the following expression:
$$\frac{x^3-27}{x^2+x-6}$$
Thanks for the help. Haven't seen this stuff since high school and I'm trying to help my younger sister out.
| The general idea would be to factor, though there aren't any common factors here:
$$\frac{x^3-27}{x^2+x-6} = \frac{(x-3)(x^2+3x+9)}{(x+3)(x-2)}$$
However, if that + in the denominator is the wrong sign, then this can be reduced, assuming $x\neq3$:
$$\frac{x^3-27}{x^2-x-6} = \frac{(x-3)(x^2+3x+9)}{(x-3)(x+2)}=\frac{x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Finding volume using triple integrals. Use a triple integral to find the volume of the solid: The solid enclosed by the cylinder $$x^2+y^2=9$$ and the planes $$y+z=5$$ and $$z=1$$
This is how I started solving the problem, but the way I was solving it lead me to 0, which is incorrect. $$\int_{-3}^3\int_{-\sqrt{9-y^2}}^... | Ok. So you have the triple integral:
$$\begin{align}
\int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\int_1^{5-y} \;dz\;dx\;dy
&= \int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}4-y\;dx\;dy \\
&=\int_{-3}^34x-xy\Bigg|_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\;dy \\
&=\int_{-3}^38\sqrt{9-y^2}-2y\sqrt{9-y^2}\;dy \\
&= 8\int_{-3}^33\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/457557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Confused about sum of G.P. formula The sum of $n$ terms of the G.P. $a, ar, ar^2, ar^3, \ldots, ar^{n-1}$ is given by $a\dfrac{r^n-1}{r-1}$.
Now consider these two progressions:
1) $r^2, r^4, r^6, ..., r^{2n}$
2) $r, r^3, r^5, ..., r^{2n-1}$
Both of these have $n$ terms. Therefore, $S_{n1} = r^2 \dfrac{(r^2)^n-1}{r-1} ... | The last term of the second series is $r^{2n-1}=r\cdot r^{2n-2}=r\cdot (r^2)^{n-1}$, and the ratio of consecutive terms is $r^2$, not $r$, so the sum formula should be
$$S_{2n}=r\cdot\frac{r^{2n}-1}{r^2-1}\;.$$
The ratio in the first series is also $r^2$, so the sum formula yields
$$S_{1n}=r^2\cdot\frac{r^{2n}-1}{r^2-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/457872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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$\frac1a+\frac1b+\frac1c=0 \implies a^2+b^2+c^2=(a+b+c)^2$? How to prove that $a^2+b^2+c^2=(a+b+c)^2$ given that $\frac1a+\frac1b+\frac1c=0$?
| Expand the right side:
$$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc.$$
From the supplementary condition we have
$$\frac{1}{a}+\frac{1}{b} + \frac{1}{c} = 0$$
Or
$$\frac{ab+ac+bc}{abc} = 0.$$
Therefore $ab+ac+bc = 0$ and the result follows. (None of $a,b,c$ can be $0$ else their inverses would be undefined and so the... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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"answer_id": 1
} |
Prove that 360 divides (a-2)(a-1)a.a.(a+1)(a+2) there's a question which asks to prove that
360 | a2(a2-1)(a2-4)
I attempted it in the following manner.
a2(a2-1)(a2-4) = (a-2)(a-1)(a)(a+1)(a+2)(a)
The first 5 terms represent the product of 5 consecutive terms. Hence,
One of them will have a factor of 5
One of them will... | We have $360 = 8 \times 9 \times 5$, so it will suffice to check that $8,9,5|a^2(a^2-1)(a^2-4)$.
For $9$, note that $a^2$ is congruent to either $1$ or $0$ modulo $3$, so both $a(a^2-1)$ and $a(a^2-4)$ are divisible by $3$, and consequently $9|a^2(a^2-1)(a^2-4)$.
Likewise, $a^2$ is congruent to either $1$ or $0$ or $4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/459334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 4
} |
How to compute $\prod\limits^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$
How to compute
$$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}\ ?$$
My Working :
$$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1}
= 1-0 = 1$$
Is it correct
| Magic answer:
Let $f(n) =\dfrac{n(n-1)}{n^2-n+1}$. Then show $f(n+1) = \dfrac{n(n+1)}{n^2+n+1}$ and thus $$\frac{f(n)}{f(n+1)} = \frac{n(n-1)(n^2+n+1)}{n(n+1)(n^2-n+1)} = \frac{n^3-1}{n^3+1}$$
(I call this a "magic answer" just because most of the other answers here give you reasons for how you would see this, while I... | {
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"url": "https://math.stackexchange.com/questions/462082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Reduction Integration Question ($\int_0^1{\frac{x^n}{\sqrt{x+1}} dx}$) What integration by parts things do I use for a reduction formula for $\int_0^1{\frac{x^n}{\sqrt{x+1}} dx}$?
I have tried many different ways of obtaining it without success.
| $$I_n=\int\frac{x^ndx}{\sqrt{x+1}}=\int\frac{x^{n-1}(1+x-1)dx}{\sqrt{x+1}} $$
$$=\int x^{n-1}\sqrt{x+1}dx-\int\frac{x^{n-1}dx}{\sqrt{x+1}}=\int x^{n-1}\sqrt{x+1}dx-I_{n-1}$$
Again, $$\int x^{n-1}\sqrt{x+1}dx= \sqrt{x+1}\frac{x^n}n-\int \frac{x^n}{2n\sqrt{x+1}}dx=\sqrt{x+1}\frac{x^n}n-\frac{I_n}{2n}$$
$$\implies I_n=\sq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Identifying the nature of the eigenvalues I wish somebody could help me in this one. We have to choose one of the $4$ options.
Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. Consider the $3 \times 3$ matrix
$$A=\begin{bmatrix}
1 & b & c \\
b & a & 0 \\
c & 0 & 1 \\
\end{bm... | Let $\mathbf{x}=[x_1\quad x_2\quad x_3]^{T}$ is an arbitrary vector in $\mathbb{R}^3$. Then,\begin{align} \mathbf{x^T}A\mathbf{x}=&[x_1\quad x_2\quad x_3]\begin{bmatrix}
1 & b & c\\
b & a & 0\\
c & 0 & 1\\
\end{bmatrix}\begin{bmatrix}
x_1\\
x_2\\
x_3
\end{bmatrix} \\
\ =& x_1^2+x_3^2+ax_2^2+2bx_1x_2+2cx_1x_3\\
\ =& (x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $\cos^4 \theta −\sin^4 \theta = x$. Find $\cos^6 \theta − \sin^6 \theta $ in terms of $x$. Given $\cos^4 \theta −\sin^4 \theta = x$ , I've to find the value of $\cos^6 \theta − \sin^6 \theta $ .
Here is what I did:
$\cos^4 \theta −\sin^4 \theta = x$.
($\cos^2 \theta −\sin^2 \theta)(\cos^2 \theta +\sin^2 \theta) ... | $$
\cos^6\theta-\sin^6\theta = \left ( \cos^2 \theta\right )^3 - \left (\sin^2 \theta \right )^3 = \\
= \left( \cos^2 \theta - \sin^2 \theta\right ) \left(\cos^4 \theta + \sin^2\theta \cos^2\theta + \sin^4\theta \right ) = \\
= x \left ( \cos^4 \theta - 2\cos^2\theta\sin^2\theta + \sin^4 \theta + 3 \cos^2\theta\sin^2\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 0
} |
Solution to cubic inequality How do I find the solutions to this equation? $$(3x^3)-(14x^2)-(5x) \leq 0$$
| Step 1. Find the roots of the cubic equation $3x^3-14x^2-5x=0$.
$$
\begin{align*}
x(3x^2-14x-5)&=0\\
x(3x+1)(x-5)&=0\\
x&=0\quad\text{or}\\
x&=-\frac{1}{3}\quad\text{or}\\
x&=5
\end{align*}
$$
Step 2. Draw a sign table for $P=3x^3-14x^2-5x$.
Step 3: From your sign table, you can see that for $x\in(-\infty,-\frac{1}{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
For $a$, $b$, $c$, $d$ the sides of a quadrilateral, show $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. (A generalization of IMO 1983 problem 6)
Let $a$, $b$, $c$, and $d$ be the lengths of the sides of a quadrilateral. Show that
$$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0 \tag{$\star$}$$
Background: The well ... | WLOG $a = \max(a,b,c,d)$
Let $t = \frac{1}{2} (b+c+d-a) \ge 0$ because $a$,$b$,$c$,$d$ are sides of a quadrilateral
Then decreasing $(a,b,c,d)$ simultaneously by $t$ reduces the desired expression
And $a-t = (b-t)+(c-t)+(d-t)$
Thus it suffices to minimize the expression when $a=b+c+d$, which reduces to:
$
\begin{align}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 4,
"answer_id": 0
} |
Inequality with square roots: $\sqrt{x^2+1}+\sqrt{y^2+1}\ge \sqrt{5}$
Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. How do I show that $\sqrt{x^2+1}+\sqrt{y^2+1}\ge \sqrt{5}$?
How do I deal with square roots inside the inequality?
| Let $f(x) = \sqrt{x^2+1}+\sqrt{(1-x)^2+1}$. As $x$ and $y$ are non-negative integers, we only consider $x \in [0, 1]$.
Note that $f(x)$ is symmetric about $x = \frac{1}{2}$ and
$$f'(x) = \frac{x\sqrt{(1-x)^2+1} + (x - 1)\sqrt{x^2+1}}{\sqrt{x^2+1}\sqrt{(1-x)^2+1}}.$$
For $0 < x < \frac{1}{2}$, $x < 1 - x$ so $\sqrt{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 0
} |
Double Integral Question on unit square Hints on solving following double integral will be appreciated.
$$\int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x$$
| \begin{align}
\frac{x^{2} - y^{2}}{(x^{2}+y^{2})^{2}}
&=
{x^{2} + y^{2} -2y^{2} \over \left(x^{2} + y^{2}\right)^{2}}
=
{1 \over \left(x^{2} + y^{2}\right)^{2}}\,\left\lbrack%
{\partial y \over \partial y}\,\left(x^{2} + y^{2}\right)
-
{\partial\left(x^{2} + y^{2}\right) \over \partial y}\,y
\right\rbrack
\\[3mm]&=
{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
proof for the Ramanujan's formula ? I found this formula in a textbook in which the proof to the formula was not given
Ramanujam's formula
$$\sqrt{1 +n\sqrt{1 +(n+1)\sqrt{1 + (n+2)\sqrt{1 + (n+3)\sqrt{1 +....\infty}}}}} = n+1$$
Its a great equation andhow do you prove this. its a bit difficult for me and tried differen... | For all $x > 0$ and $m \in \mathbb{N}$, define $\varphi_m(x)$ by:
$$(x+1)\varphi_m(x) = \begin{cases}1,&\text{ for } m = 0\\
\\
\sqrt{1+x\sqrt{1+(x+1)\sqrt{\cdots\sqrt{1+(x+m)}}}},&\text{ otherwise. }\end{cases}$$
It is clear for $m > 0$, these functions satisfy the recurrence relations:
$$\begin{align}
&(x+1)^2\v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Probability in coin toss Suppose that you have a fair coin. You start with $\$0$. You win $\$1$ each time you get a head and loose $\$1$ each time you get tails. Calculate the probability of getting $\$2$ without getting below $\$0$ at any time.
| Consider $P_i$ the probability to obtain $\$2$ when you have $\$i$, then we look for $P_0$. So a recurrence can be
$$P_{i}=\frac{1}{2}P_{i-1} + \frac{1}{2}P_{i+1}$$
with $P_2=1$ and $P_{-1}=0$ because we do not allow a score below of $\$0$. Now
$$P_0=\frac{1}{2}P_{-1} + \frac{1}{2}P_{1}=\frac{1}{2}P_{1},$$
$$P_1=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ $a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$
I have tried to use AM-GM inequality, but get no result as follows:
$$a+\sqrt{ab}+\sqrt[3]{abc}\leq a+\frac{a+b}{2}+\frac{a+b+c}{3}$$
| $a+\sqrt{ab}+\sqrt[3]{abc}=a+\frac{\sqrt{4ab}}{2}+\frac{\sqrt[3]{64abc}}{4}\le a+\frac{a+4b}{4}+\frac{a+4b+16c}{12}=\frac{4(a+b+c)}{3}=28$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/477283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Is it possible to put $+$ or $-$ signs in such a way that $\pm 1 \pm 2 \pm \cdots \pm 100 = 101$? I'm reading a book about combinatorics. Even though the book is about combinatorics there is a problem in the book that I can think of no solutions to it except by using number theory.
Problem: Is it possible to put $+$ or... | Replacing 100 with $n$
and using Brian M. Scott's solution,
we want a partition of
$\{1, 2, ..., n+1\}$
into two sets with equal sums.
The sum is
$\frac{(n+1)(n+2)}{2}$,
and if $n=4k$,
this is
$(4k+1)(2k+1)$
which is odd
and therefore impossible.
If $n = 4k+1$,
this is
$(2k+1)(4k+3)$
which is also odd,
and therefore im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 5,
"answer_id": 2
} |
Sum of the series $1+\frac{1\cdot 3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+\cdots$ Decide if the sum of the series
$$1+\frac{1\cdot3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+ \cdots$$
is: (i) $\infty$, (ii) $1$, (iii) $2$, (iv) $4$.
|
O method of differences, so powerful and yet so despised...
The $n$th term of the series to be computed is
$$
\prod_{k=1}^n\frac{2k+1}{2k+4}=\frac4{2n+4}\prod_{k=1}^n\frac{2k+1}{2k+2}=4(1-a_{n+1})\prod_{k=1}^na_k
$$ where $$a_k=\frac{2k+1}{2k+2}$$
By telescoping, each partial sum of the series is
$$
\sum_{n=0}^{N-1}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/479610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 8,
"answer_id": 3
} |
The number of solutions to $\frac{1}x+\frac{1}y+\frac{1}z=\frac{3}n,x,y,z\in\mathbb N$ Denote
$$g(n)=\{\{x,y,z\}\mid \frac{1}x+\frac{1}y+\frac{1}z=\frac{3}n,x,y,z\in\mathbb N\},$$
$$h(n)=\{\{x,y,z\}\mid \frac{1}x+\frac{1}y+\frac{1}z=\frac{3}n,1\leq x\leq y\leq z,x,y,z\in\mathbb N\},$$
let $f(n)=|g(n)|$ be the number ... | Since I already know how to prove them, I write a proof here now.
It's easy to see that in $h(n)$,
(1)if $x,y,z$ are distinct, then $x,y,z$ add $6$ to $f(n)$,
(2)if just two of them are equal, add $3$ to $f(n)$,
(3)if $x=y=z$, then they add $1$ to $f(n)$.
Since $6$ is even, case (1) didn't change the parity of $f(n)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/481049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to prove :$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2...u_n}\right)^2=2011$ For sequence $u_n$ satisfing : $$\begin{cases} u_1=\sqrt{2015}\\ u_{n+1}=u_n^2-2\end{cases}$$
How to prove : $$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2...u_n}\right)^2=2011$$
| Here I use its properties as an additive telescoping series:
$$\left(\frac{u_n^2-2}{u_1\cdots u_n}\right)^2=\frac{u_n^4}{(u_1\cdots u_n)^2}-\frac{4u_n^2}{(u_1\cdots u_n)^2}+\frac{4}{(u_1\cdots u_n)^2}$$
$$=\frac{u_n^2}{(u_1\cdots u_{n-1})^2}-\frac{4}{(u_1\cdots u_{n-1})^2}+\frac{4}{(u_1\cdots u_n)^2}$$
Substitute again... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/481900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 0
} |
Compute $ \int_{-\infty}^{\infty} \frac{x^2}{(1+x^2)^2} dx$ Compute $$ \int_{-\infty}^{\infty} \frac{x^2}{(1+x^2)^2} dx$$
Of course we have $$ \int_{-\infty}^{\infty} \frac{x^2}{(1+x^2)^2} dx = 2 \int_{0}^{\infty} \frac{x^2}{(1+x^2)^2} dx = 2 \int_{0}^{\infty} \left( \frac{x}{1+x^2} \right) ^2 dx = \lim_{ A \to \inft... | We can use Parseval's theorem!
If $F$ is the Fourier transform of $f$, and $G$ is the Fourier transform of $g$, then
$$
\int_{-\infty}^\infty\overline{f(t)}g(t)\,dt=
\frac{1}{2\pi}\int_{-\infty}^\infty\overline{F(x)}G(x)\,dx.
$$
We have the Fourier transform pairs
*
*$\displaystyle e^{-|t|}\longmapsto\frac{2}{1+\ome... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/483180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 6
} |
Solving a system of equations using modular arithmetic modulo 5 Give the solution to the following system of equations using modular arithmetic modulo 5:
$4x + 3y = 0 \pmod{5}$
$2x + y \equiv 3 \pmod{5}$
I multiplied $2x + y \equiv 3 \pmod 5$ by $-2$, getting $-4x - 2y \equiv -6 \pmod{5}$.
$-6 \pmod{5} \equiv 4 \... | Sign error on substitution, it should be $x\equiv -3\pmod{5}$.
You had $4x+(3)(4)\equiv 0$, that is, $4(x+3)\equiv 0$. From this we get $x+3\equiv 0$, so $x\equiv -3\pmod{4}$.
Negative numbers are sometimes troublesome, so we may wish to rewrite as $x\equiv 2\pmod{5}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/484467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Computing Infinite Continued Fractions I am looking for "tricks" used to compute infinite continued fractions.
For example, $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$$ is the golden ratio since if we denote it by $x$, then we have $$x=1+\frac{1}{x},$$ which simplifies to $$x^2-x-1=0$$
Are there any other (differ... | This Infinite Continued Fractions can be written in many different way.
$$ \frac{1}{1} ;\frac{1}{1+\frac{1}{1}};\frac{1}{1+ \frac{1}{1+\frac{1}{1}}};\frac{1}{1+ \frac{1}{1+ \frac{1}{1+\frac{1}{1}}}} ... $$
like this:
$$ \frac{1}{1} ;\frac{1}{2};\frac{3}{2};\frac{5}{3};\frac{8}{5};\frac{13}{8};\frac{21}{13}..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Olympiad number theory problem I found this problem in previous problems of the olympiads of my country
If $t^2+n^2=r^2$, where $t$ has $3$ positive divisors, $n$ has $30$ positive divisors and $t,n,r$ are natural numbers, find the sum of all the possible values of $t$
I did gave it a try but only solved it partiall... | Let us start as you did. I will assume that $t$, $n$ and $r$ are coprime. Then $t=a^2-b^2$ and $n=2ab$. See the answer of André Nicolas for the case when $t$, $n$ and $r$ are not coprime.
We get that
\begin{align*}
p^2 &= 2b+1,\\
n &= 2(b+1)b.
\end{align*}
Therefore,
$$n = \frac{p^4-1}{2} = (p-1) \times (p+1) \times \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
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Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$ If $a, b$ are positive real numbers and $a+b = 1$, prove that :
$$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \geq \frac{25}{2}$$
I can see that the value $\frac{25}2$ is attained for $a=b=\frac12$. But I do not kno... | We can use the inequalities between quadratic mean (a.k.a. square root mean), arithmetic mean, geometric mean and harmonic mean
$$\sqrt{\frac{x^2+y^2}2} \ge \frac{x+y}2 \ge \sqrt{xy} \ge \frac2{\frac1x+\frac1y}.$$
These inequalities are true for any $x,y>0$. The equality holds if and only if $x=y$. They can be generali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 10,
"answer_id": 1
} |
Sum $ \sum\limits_{k=1}^{n} (-1)^k \frac{2k+3}{k(k+1)} $ I have the following Sum
$$ \sum\limits_{n=1}^{\infty} (-1)^n \frac{2n+3}{n(n+1)} $$
and I need to calculate the sums value by creating the partial sums.
I started by checking if $$\sum\limits_{n=1}^{\infty} \left| (-1)^n \frac{2n+3}{n(n+1)} \right|$$ converge... | \begin{align}
?
&\equiv
\sum\limits_{n = 1}^{\infty}\left(-1\right)^n \frac{2n + 3}{n\left(n + 1\right)}
=
\sum_{n = 1}^{\infty}\left\lbrack%
{4n + 3 \over 2n\left(2n + 1\right)}
-
{4n + 1 \over \left(2n - 1\right)\left(2n\right)}
\right\rbrack
\\[3mm]&=
\sum_{n = 1}^{\infty}\left\lbrack%
{1 \over n}
+
{1 \over 2n\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
drawing balls from an urn (conditional probability) Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ... |
The answer is the Probability that the first ball is red and the second ball is black, divided by the Probability that the second ball
is black.
$\frac{P(\text{First Ball is RED AND Second Ball is BLACK) }}{P(\text{Second Ball is BLACK) }}$
P(1st ball = Red and 2nd ball = Black )
$=\frac{1}{2}[ (\frac{2}{6}* \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/488937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Bessel function to $\sin(kr)$ $J_{\frac{1}{2}}(kr)=\frac{\sqrt{\frac{2}{\pi }} \text{Sin}[\text{kr}]}{\sqrt{\text{kr}}})$ This can be easily obtained by Mathematica,
How to do the details?
| One way is to use the series definition of the Bessel function together with the duplication formula for the Gamma function.
For $x \geq 0$ we have
$$
\begin{align}
J_{1/2}(x) &= \sqrt{\frac{x}{2}} \sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1+1/2)} \left(\frac{x}{2}\right)^{2n} \\
&= \sqrt{\frac{2}{x}} \sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/489582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find all positive integers $n$ for which $1 + 5a_n.a_{n + 1}$ is a perfect square. The sequence $a_1, a_2, \ldots $ is defined by the initial conditions $$a_1 = 20; \quad a_2 = 30$$ and the recursion
$$a_{n+2} = 3a_{n+1} - a_n$$ and
for $n \geq 1$. Find all positive integers $n$ for which $1 + 5a_n * a_{n+1}$ is a pe... | The only such $n$ is $n=3$, with
$$
1 + 5 a_3 a_4 = 1 + 5 \cdot 70 \cdot 180 = 63001 = 251^2.
$$
Let $b_n = a_n/10 = 2, 3, 7, 18, 47, \ldots$ for $n=1,2,3,4,5,\ldots$ .
These are sums of consecutive odd-order Fibonacci numbers:
$2 = 1+1$ (with the first $1$ being $F_{-1}$),
$3 = 1+2$, $7 = 2+5$, $18 = 5+13$, $47 = 13+3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/489663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
find this limit $\lim_{x\to0^{+}}\frac{\tan{(\tan{x})}-\tan{(\sin{x})}}{\tan{x}-\sin{x}}$ find the limit.
$$\lim_{x\to0^{+}}\dfrac{\tan{(\tan{x})}-\tan{(\sin{x})}}{\tan{x}-\sin{x}}$$
my try:
$$\tan{x}=x+\dfrac{1}{3}x^3+o(x^3),\sin{x}=x-\dfrac{1}{6}x^3+o(x^3)$$
so
$$\tan{(\tan{x})}=\tan{x}+\dfrac{1}{3}(\tan{x})^3+o(\t... | Factor a $\tan(x)$ from the denominator:
$$\frac{\tan\tan(x) - \tan\sin(x)}{\tan(x)-\sin(x)} = \frac{\frac{\tan\tan(x)}{\tan(x)} - \cos(x)\frac{\tan\sin(x)}{\sin(x)}}{1-\cos(x)}.$$
The quotients in the numerator go to $1$, so the numerator goes to $1-\cos(x)$. This suggests cancellation.
After two derivatives, the deno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/490470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Prove $3^n \ge n^3$ by induction Yep, prove $3^n \ge n^3$, $n \in \mathbb{N}$.
I can do this myself, but can't figure out any kind of "beautiful" way to do it.
The way I do it is:
Assume $3^n \ge n^3$
Now,
$(n+1)^3 = n^3 + 3n^2 + 3n + 1$,
and $\forall{} n \ge 3$,
$3n^2 \le n^3, \,\, 3n + 1 \le n^3$
Which finally giv... | Here is another argument, but it's not necessarily simpler than yours:
1) If $n=1$, $3^1=3\ge1=1^3$; if $n=2$, $3^2=9\ge8=2^3$; and if $n=3$, $3^3=3^3$.
2) Now assume that $3^n\ge n^3$ for some integer $n\ge3$.
Then $\displaystyle\frac{1}{n}\le\frac{1}{3}$, so $\displaystyle\frac{(n+1)^3}{n^3}=\big(1+\frac{1}{n}\big)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/490983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Contest math problem algebra proof Let $r, s$ be integers and let
$$a = (2011)^2 + (2011)r + s$$ and
$$b = (2012)^2 + (2012)r + s$$
Show that there exists an integer $c$ with $c^2 + rc + s = ab$.
Can anyone help me with this?
| Choose r = 2 and s = 1
Then $2011^2 + 2.2011 + 1 = a$
and $2012^2 + 2.2012 + 1 = b$
$a = (2011 + 1)^2$
$b = (2012 + 1)^2$
$ab = (2012.2013)^2$
Let 2012 = u
$ab = (u(u+1))^2$
$ab = u^4 + 2u^3 + u^2$
Let $c = au^2+bu+l$
$ab = (au^2+bu+l)^2 + (au^2 + bu +l)*2 + 1$
$a^2u^4 + b^2u^2+l^2+2(au^2bu+2bul+alu^2)+2au^2+2bu+2c+1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
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