AsdrubalBeltran
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4 answers
6 votes
522 views
Is the reasoning/algebra for my proof correct? (musical tuning theory proof)
14 votes

Continuing with the idea, of @sanath: The second step asserts that $3^m=2^{m-n}$. Here, $2$ and $3$ are prime numbers, so this equation then implies that there is a natural number with two different ...

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4 answers
4 votes
771 views
Integrating $\sin(y^2)$
11 votes

Hint: make $u=y^2$ then $du=2y\,dy$

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1 answers
4 votes
19k views
Derivative of Integral with variable bounds
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11 votes

Yes is correct, remember that $$\frac{d}{dx}\int_{g(x)}^{f(x)}h(t)\,dt=h(f(x))\cdot f'(x)-h(g(x))\cdot g'(x) $$ this is by the second theorem of calculus and by chain rule.

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3 answers
3 votes
943 views
How do I get rid of the radical in the denominator of $\lim\limits_{x \to -9} \frac{x+9}{\sqrt{x+9}}$?
10 votes

$$\frac{x+9}{\sqrt{x+9}}=\frac{(x+9)^1}{(x+9)^{\frac{1}{2}}}=(x+9)^{1-\frac{1}{2}}=(x+9)^{\frac{1}{2}}$$

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3 answers
3 votes
473 views
Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
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9 votes

Hint: Note that $2|a|=|a+b+a-b|\leq|a+b|+|a-b|$ $2|b|=|b+a+b-a|\leq|b+a|+|b-a|=|a+b|+|a-b|$ then $$2|a|+2|b|\leq 2(|a+b|+|a-b|)$$ then $$|a|+|b|\leq |a+b|+|a-b|$$

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2 answers
3 votes
188 views
What property was used in this sine transformation?
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7 votes

The identity is $$\sin^3x=\frac{3\sin x-\sin(3x)}{4}$$ Proof: \begin{align*} 4\sin^3x-3\sin x &= \sin x(4\sin^2x-3) \\ &=\sin x(1-4\cos^2x) \quad \text{Pythagorean identity} \\ &...

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4 answers
3 votes
372 views
Formula for $1! \times 2! \times \cdots \times n!$?
7 votes

A posibble form is: $$\prod_{i=1}^ni!$$ In other way is $$1^n\cdot2^{n-1}\cdot3^{n-2}\cdot\cdots\cdot(n-1)^2\cdot n^1$$

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6 answers
3 votes
4k views
Prove that $\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$
7 votes

If $y=x$ the limit is $0$, but if $x=y^2$ the limit is $\frac{1}{2}$, then the limit don't exist.

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1 answers
2 votes
222 views
Ramanujan's Nested Radicals: evaluating $\sqrt{4+\sqrt{16+\sqrt{64+\sqrt{\cdots}}}}$
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6 votes

The mistake is: $2\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{\cdots}}}} \neq \sqrt{4^2 + \sqrt{4^3 + \sqrt{4^4 + \sqrt{\cdots}}}} $ $2\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{\cdots}}}} = \sqrt{4^2 + \sqrt{4^4 + \...

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6 answers
4 votes
390 views
Limit of sequences: $\lim \frac{(2n)!}{(n!)^2} $
6 votes

Hint: Note that $$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}$$ Then the sequence diverges.

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6 answers
9 votes
582 views
Prove that $\lim_{x\rightarrow 1}{\frac{x^n-1}{x-1}}=n$ for all integer n without L'Hôpital
6 votes

Note that if $f(x)=x^n$ then $$f'(1)=\lim_{x\to 1}\frac{x^n-1}{x-1}$$ How $$f'(x)=nx^{n-1}$$ then $$f'(1)=n\cdot(1^{n-1})=n$$

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3 answers
3 votes
98 views
If $ab + c = 1993,$ what is $a + b + c?$
5 votes

You can make cases $ab=1993-2=1991=11\cdot181$ $ab=1993-3=1990=2\cdot5\cdot199$ $ab=1993-5=1988=4\cdot7\cdot71$ $ab=1993-7=1986=2\cdot3\cdot331$

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2 answers
3 votes
177 views
$ x \ge 0\text{ and } y \ge 0 \implies \frac{x+y}{2} \ge \sqrt{xy} $
5 votes

Note that: $$(x-y)^2\ge 0\implies x^2+y^2\ge2xy\implies x^2+2xy+y^2\ge4xy\implies(x+y)^2\ge4xy$$

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5 answers
9 votes
307 views
Show $7!^{1/7} < 8!^{1/8}$
5 votes

Hint: Note that $$7!<8^7$$ then $$7!<8^7\implies(7!)^\frac{1}{7}<8\implies7!\cdot(7!)^\frac{1}{7}<7!\cdot8\implies(7!)^\frac{8}{7}<8!\implies(7!)^\frac{1}{7}<(8!)^\frac{1}{8}$$

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2 answers
5 votes
572 views
$a+b+c+d+e$ divides $a^5+b^5+c^5+d^5+e^5-5abcde$
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5 votes

Hint: firts note that; if $a(b+c)+b(c+d)+c(d+e)+d(e+a)+e(a+b)=0$, then $$(a+b+c+d+e)^2=a^2+b^2+c^2+d^2+e^2$$ Now take $P(x)=x^5+kx^4+rx^3+sx^2+tx+u$, with roots $a,b,c,d,e$ then from Viète’s ...

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3 answers
5 votes
210 views
Show that $0=x^x$ has no solution in $\mathbb{R}$.
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5 votes

If Supposed that there is $t^t=0$, with $t>0$, if $t>0$ then $$\ln{t}\in \mathbb{R}$$ $$t\cdot\ln{t}\in \mathbb{R}$$ $$\ln{t^t}\in \mathbb{R}$$ but $$\ln{t^t}=\ln{0}\in \mathbb{R}$$ is a ...

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1 answers
6 votes
197 views
What's a good introductory text to ZF set theory?
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4 votes

I like Introduction to set theory

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2 answers
2 votes
138 views
How to prove this equation? $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = \frac{1}{a+b+c}$
4 votes

Note that: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = \frac{1}{a+b+c}$$ $$\frac{ab+ac+bc}{abc}=\frac{1}{a+b+c}$$ $$(ab+ac+bc)(a+b+c)-abc=0$$ $$(a+b)(b+c)(c+a)=0$$ then $a=-b$, $a=-c$ or $b=-c$. If $a=-...

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2 answers
1 votes
2k views
Why does the sequence ${1/n}$ not converge in the positive reals?
4 votes

the definition of the rudin book: "A sequence $\{p_n\}$ in a metric space $X$ is said to converge if there is a point $**p \in X**$ with the following property...$$\lim_{n\to\infty }\{p_n\}=p$$ how $...

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10 answers
5 votes
2k views
Number theory problem book
4 votes

Number Theory: Structures, Examples, and Problems Titu Andreescu, Dorin Andrica Problem-Solving and Selected Topics in Number Theory, by Michael Th. Rassias

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5 answers
3 votes
14k views
Let A be a set of all infinite sequences consisting of 0's and 1's. Prove that A is not countable.
3 votes

This is equivalent to proof that: $|\{0,1\}^{\mathbb{N}}|=|P(\mathbb{N})|$ You can use that if $A⊆\mathbb{N}$, then $f(a)=1$ if $a∈A$ otherwise $f(b)=0$, then $$f:P(A)\to\{0,1\}^{\mathbb{N}}$$ is a ...

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6 answers
-2 votes
72 views
Prove that there is no real number L such that $|1−L| < 1/2$ and $ |1+L| < 1/2$
3 votes

Note that: $$2=|1-L+1+L|\leq|1-L|+|1+L|<1$$

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2 answers
2 votes
66 views
Let $\frac{\tan A}{1-\tan^2A}=\sin^220^\circ-\sin160^\circ\sin220^\circ+\sin^2320^\circ$, find $\tan6A$
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3 votes

Note that: \begin{array}{} \dfrac{\tan2A}{2}&=&\sin^2160^\circ-\sin160^\circ\sin220^\circ+\sin^2220^\circ\\ \dfrac{\tan2A}{2}&=&(\sin160^\circ-\sin220^\circ)^2+\sin160^\circ\sin220^\...

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3 answers
0 votes
55 views
solve an irrational equation $1+\frac{\sqrt{2x^2+1}}{|x|}=(x^2+x)(1+\sqrt{x^2+2x+3})$
3 votes

An a way is:$$1+\frac{\sqrt{2x^2+1}}{|x|}=(x^2+x-1+1)(1+\sqrt{x^2+2x+3})$$ $$1+\frac{\sqrt{2x^2+1}}{|x|}=(x^2+x-1)(1+\sqrt{x^2+2x+3})+1+\sqrt{x^2+2x+3}$$ $$\sqrt{2x^2+1}-|x|\sqrt{x^2+2x+3}=|x|(x^2+x-1)...

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2 answers
7 votes
170 views
why $\left(\left( \left(-\frac{1}{4}\right)^{-2}\right)^\frac{1}{4}\right) \neq \left(\left(-\frac{1}{4}\right)^{-\frac{1}{2}}\right)$?
3 votes

Note that $\sqrt[n]{x^n}=|x|$ with $n$ even: $$(((-4)^2)^{1/2})^{1/2}=(|-4|)^{1/2}=2$$

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2 answers
2 votes
47 views
Find $\underset{x \to 2} \lim \frac{-4x+8}{|x-2|}$?
3 votes

Note that if $$x\to 2^+\implies |x-2|=x-2$$ and if $$x\to 2^-\implies |x-2|=-(x-2)$$

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2 answers
2 votes
341 views
If $p,q,r$ be three positive numbers, then prove that the value of $(p+q)(q+r)(r+p) \ge 8pqr$
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3 votes

Note that $$r (p - q)^2 + p (q - r)^2 + q (r - p)^2\geq0$$ And $$r (p - q)^2 + p (q - r)^2 + q (r - p)^2=(p+q)(q+r)(p+r)-8pqr$$

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3 answers
1 votes
372 views
Mathematical Induction divisibility $8\mid 3^{2n}-1$
3 votes

Hint: Note that: $$3^{2(n+1)} -1=9\cdot3^{2n}-1=(3^{2n}-1)+8\cdot3^{2n}$$

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1 answers
2 votes
89 views
Existence of roots of $A_1\sin(\omega_1t+\phi_1)+A_2\sin(\omega_2t+\phi_2)$
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3 votes

Hint: Suppose that $A_1=\max\{A_1,A_2\}$, then take $t_0$, such that $\omega_1t_0+\phi_1=\frac{\pi}{2}$ and take $t_1$, such that $\omega_1t_1+\phi_1=\frac{3\pi}{2}$, then: $$f(t_0)>0$$ and $$f(...

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1 answers
4 votes
2k views
Bijection between natural numbers $\mathbb{N}$ and natural plane $\mathbb{N} \times \mathbb{N}$
3 votes

A bijection is: $$f(m,n)=2^m(2n+1)-1$$

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