JavaMan
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Number of ways of reaching a point from origin
Accepted answer
11 votes

It is well known that the number of ways to get to the lattice point $(x,y)$ (supposing $x, y \geq 0$) by taking steps of one unit each either in the eastward or northward direction is exactly $$ {x +...

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Which of the numbers $99^{100}$ and $100^{99}$ is the larger one?
64 votes

Note that $$\begin{align} 99^{100} > 100^{99} &\iff 99 \cdot 99^{99} > 100^{99} \\ &\iff 99 > (100/99)^{99} \\ &\iff 99 > \left( 1 + \frac{1}{99}\right)^{99} \end{align}$...

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recursive relation with sequences
Accepted answer
0 votes

Hint: $$\begin{align} t_1 &= 1 \\ t_2 &= 1 + 2 \\ t_3 &= 1 + 2 + 3 \\ t_4 &= 1 + 2 + 3 + 4 \end{align}$$ Can you guess a formula for $t_n$? Once you have a formula for $t_n$,...

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convergence of a trigonometric series
1 votes

Hint: $$\begin{array} &\sin(\pi) = 0 \\ \sin(2 \pi) = 0 \\ \sin(3\pi) = 0 \\ \qquad\qquad\vdots \end{array}$$

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Is there a formula for $(1+i)^n+(1-i)^n$?
6 votes

This is not a particularly elegant solution, but an alternative route is to simply note that $(1 \pm i)^2 = \pm 2i$. I will show a lot of steps, but this method involves only very easy calculations. ...

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How many solutions has $z^\pi = 1$?
Accepted answer
6 votes

The number of solutions is always infinity or a nonnegative integer. Let $z = re^{i \theta}$, to see that $$ z^{\pi} = r^{\pi}e^{i \pi\theta} = 1. $$ The solutions are then $r = 1$ and $\theta = ...

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convergence of sum of series
Accepted answer
2 votes

Hint: $$\begin{align} \sin(x) = &x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 + \dots. \\ &\Downarrow \\ \frac{\sin(x)}{x} = &1 - \frac{1}{3!}x^2 + \frac{1}{5!}x^4 + \dots \\ &\...

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Slightly confused about Möbius' inversion theorem
1 votes

Euler showed that the Dirichlet series of a multiplicative function $f$ can be written as a product (now called an Euler product): $$ L(s, f) := \sum_{n \geq 1} \frac{f(n)}{n^s} = \prod_p \left(1 + ...

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Solving $z^4 + 2z^3 + 6z - 9 = 0$
2 votes

The complex number $z = c$ is a solution to the equation $f(z) = 0$ (also called a zero or root of $f(z)$) if and only if $(z-c)$ is a factor of $f(z)$. The proof of this statement follows easily by ...

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Intuition around why domain of x of arcsine and arccosine is [-1;1] for "real result" & domain for arctangent is all real numbers
1 votes

If $f(x)$ and $f^{-1}(x)$ are inverse functions (meaning that $(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$ on the respective domains), then the domain of $f$ is the range of $f^{-1}$ and the domain ...

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Proving $\mathbb{N}^k$ is countable
6 votes

Hint: It is enough to prove that $A \times B$ is countable whenever $A$ and $B$ are countable. To do this, write $$ A = \{a_0,a_1,a_2, \dots\} $$ and $$ B = \{b_0,b_1,b_2,\dots\}. $$ To find a ...

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How would I know if $f(x)=x^5-2x+10$ has a root at the interval $[-2, 2]$?
3 votes

Hint: $$f(-2) = (-2)^5 -2(-2) + 10 = -32 + 4 + 10 = -18 < 0$$ while $$f(2) = 2^5 - 2(2) + 10 = 38 > 0.$$

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Is the cross-product of two displacement vectors orthogonal to both of them?
2 votes

The Cross Product $v \times w$ is always orthogonal to both $v$ and $w$. This is easy to see by a direct calculation: Write $v = \langle v_1, v_2 , v_3\rangle$ and $w = \langle w_1, w_2, w_3\rangle ...

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Determining the Value of a Gauss Sum.
Accepted answer
4 votes

They do not provide a derivation, but this is actually written up in Wikipedia. I use the standard notation $e(x) = \exp(2 \pi i x)$. Assuming $\gcd(k,n) = 1$, we have $$ \sum_{x \in \mathbb{Z}/n \...

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Prove a property of divisor function
1 votes

Hint $\# 2$: tards answer might be a simpler way to go, but here is an alternative route: Let $d(n)$ denote the number of divisors of the integer $n$. Show that $$ \gcd(m,n) = 1 \quad \implies \...

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A Rough Estimation for the number of square free integers
5 votes

Rough sketch: Have you seen the Wikipedia section on the distribution of squarefree integers? It essentially gives your solution. All you need to notice is that for large $n$, the "probability" ...

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Prove how many distinct elements in the set $\{ax \pmod{m}:a\in\{0,...,m-1\}\}$
1 votes

If $\gcd(x,m) = d > 1$, then the equation $y \equiv ax \pmod{m}$ has $d$ times the solutions of $y \equiv a\frac{x}{d} \equiv \frac{m}{d}$. In other words, we may assume $\gcd(x,m) = 1$. In ...

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How can I prove that the binomial coefficient ${n \choose k}$ is monotonically nondecreasing for $n \ge k$?
3 votes

Fix $k \geq 0$. Show that $${n+1 \choose k} = \frac{(n+1)!}{k!(n+1-k)!} \geq \frac{n!}{k!(n-k)!} = {n \choose k}.$$ But this follows whenever $n+1 \geq n+1 - k$, so...

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RSA: How Euler's Theorem is used?
4 votes

Aside from it being unlikely that $\gcd(x,n) \neq 1$, note that the only possibilities are $\gcd(x,n) \in \{1,p,q,n\}$. Therefore, if $x$ and $n$ are not coprime, the one can decipher the text ...

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Proving $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n}$ holds by induction
1 votes

For the Inductive Step: Suppose that we have $$ \frac{n^n}{3^n} < n! $$ and $$ n! < \frac{n^n}{2^n}. $$ We want to show that this implies that $$ \frac{(n+1)^{n+1}}{3^{n+1}} < (n+1)! ...

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Sum of two countably infinite sets
Accepted answer
7 votes

Hint: Assuming "sum" means "union", let $A = \{a_1, a_2, \dots\}$ and $B = \{b_1 , b_2, \dots\}$ be two countable sets. Consider the map $$ f(n) = \left\{ \begin{array}{ccc} b_{n/2} & & n ...

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Two questions related to probability theory and pedagogy
0 votes

You are right. The easiest way to see this is to recognize the probability that you roll a $1$ or a $2$ as $$ 1 - Pr(\text{not rolling a }1 \text{ or } 2) = 1 - (4/6)^2 = 5/9. $$

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An identity involving the Möbius function
4 votes

Let $D_f(s)$ denote the Dirichlet generating series: $$ D_f(s) = \sum_{n =1}^{\infty} \frac{f(n)}{n^s}. $$ When the series convergences absolutely, we can write $D_f(s)$ as an Euler product: $$ ...

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paradoxical answers using 'i'
1 votes

First and foremost, $\sqrt{1}$ is not $1$ or $-1$. When we discuss the square root of a number $\sqrt{n}$ this denotes the positive or principal square root. Therefore, $\sqrt{1} = 1$ and $1$ alone. ...

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limit of the integral of function evaluated at $x^n$
2 votes

Since $\xi \in [0,1)$, then $\xi^n \to 0$. Hence, $f(\xi^n) \to f(0)$ as $f$ is continuous. An alternative proof: Since $f$ is continuous on $[0,1]$, then $f$ is uniformly continuous on $[0,1]$. ...

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Is $\sqrt{|xy|}$ equal to $\sqrt[4]{x^2y^2}$?
Accepted answer
2 votes

They are indeed equal. This follows since $$\begin{align} \sqrt[n]{x^n} = \left\{ \begin{array}{ccc} |x| & & n \text{ is even} \\ x & & n \text{ is odd} \end{array} \right. \...

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Asymptotic formula for $\sum_{n \le x} \frac{\varphi(n)}{n^2}$
7 votes

You were indeed almost there. All that's left to do is just switch the order of summation on the last sum. I won't fill in all the details but here's a start: $$\begin{align} \sum_{1 \leq n \leq x}...

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Taking the derivative of $\frac1{x} - \frac1{e^x-1}$ using the definition
2 votes

This is just an exercise in persistence. First note that $$ \lim_{h \to 0}\frac{\frac{1}{h} - \frac{1}{e^h - 1} - \frac{1}{2}}{h} = \lim_{h \to 0} \frac{2(e^h - 1)-2h - h(e^h - 1)}{2h^2 (e^h - 1)} ...

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Finding the slope of the tangent line to $\frac{8}{\sqrt{4+3x}}$ at $(4,2)$
4 votes

Just note that $$\begin{align} \frac{8}{\sqrt{4 + 3x}} - 2 &= \frac{8 - 2 \sqrt{4 + 3x}}{\sqrt{4 + 3x}} \\ &= \frac{8 - 2 \sqrt{4 + 3x}}{\sqrt{4 + 3x}} \cdot \frac{8 + 2\sqrt{4 + 3x}}{8 + ...

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Euclidean division remainder bound
0 votes

You can explicitly take $q = \left\| \frac{m}{n}\right\|$ and $r = m - n \left\| \frac{m}{n}\right\|$, where $\| \alpha \|$ denotes the nearest integer to $\alpha \in \mathbb{R}$. Note that for any ...

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