Mark Pineau
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Is it possible when multiplying two polynomials that, after collecting similar terms, all terms vanish?
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16 votes

Consider the zero polynomial $p(x)=0$, then certainly multiplying any real polynomial by $p(x)$ yields $0$. For a polynomial with $\deg\geq0$, over some field $\mathbb{F}$, this is not possible. ...

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Integrate: $\int (x^2+a^2)^{-3/2} \cdot dx$
5 votes

Here is the correct solution. Note for anything of the form $({x^2+a^2)^{n/2}}$ for n odd, consider using the following trigonometric substituion. So, let $x=atan(u) \ (or \ x=asinh(u)$ works aswell$)...

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Cauchy sequence of real numbers
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4 votes

For a sequence $\left\{s_n\right\}_{n=1}^{\infty}$ to be Cauchy, then: $\forall \epsilon > 0, \exists$ a cut off point $N\in \mathbb{N} \ s.t.$ for $m>n$, $$m,n>N\Rightarrow |s_m-s_n|<\...

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How does $e^{-j\pi n}$ become $(-1)^n$
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4 votes

Consider the power series of $e^x$, that is: $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}=1+\frac x{1!}+\frac{x^2}{2!} +\frac{x^3}{3!}+\frac{x^4}{4!}+\ ...$$ Now consider Euler's constant $e$ raised to ...

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Find the value of $(\log_{a}b + 1)(\log_{b}c + 1)(\log_{c}a+1)$ if $\log_{b}a+\log_{c}b+\log_{a}c=13$ and $\log_{a}b+\log_{b}c+\log_{c}a=8$
4 votes

We have the following: $$(log_ab+1)(log_bc+1)(log_ca+1)=(\frac {logb}{loga}+1)(\frac {logc}{logb}+1)(\frac {loga}{logc}+1)$$ Expanding yields: $$(\frac {logc}{loga}+\frac {logb}{loga}+\frac {logc}{...

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Proving $H$ is a subgroup of $GL(2,\Bbb{R})$
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3 votes

I haven't used matrices for a while in latex and apologize for any bad formatting. Hopefully someone kind enough can fix it. Also, I am assuming the operation defined on $G$, and thus $H$ is the usual ...

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Solve inequality logarithm
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3 votes

Assuming you want to show for what $x, \ log_{(1-|x|)}|(3x-1)|<1$ (with base 1-|x|), we have the following: What is the inverse of $log|x|$? (i.e. its exponential form) We also assume $logx$ has ...

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proving one lim with multiplied terms is two lims of the terms multiplied
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2 votes

It is sufficient to consider the sequence case: Consider two sequences, $\left\{a_n\right\}_{n=1}^{\infty}$ and $\left\{b_n\right\}_{n=1}^{\infty}$, such that: $$\lim_{n\to\infty}a_n=L \ and \ \lim_{...

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Math Subject exam 9768 Q.4
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2 votes

We can directly evaluate each integral and solve for the unknown constant $b$. The following is given: $$\int_0^bxdx=\int_0^bx^2dx$$ So: $$\int_0^bxdx=\bigg[\frac {x^2}{2}\bigg]^b_0=\frac {b^2}{2}$...

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Area of a circle, using $2$ triangular pyramids?
2 votes

$\mathbf{Surface \ Area \ of \ Circle:}$ By infinitely partitioning a geometric circle of radius $r$ into symmetrical sectors, and laying each unit against each other in an approximate rectangular ...

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understandin the term logarithm and log.
2 votes

This explanation will contain both a small rigorous and elementary explanation of your following query. $\mathbf{Rigerous \ Explanation:}$ The logarithmic function, denoted as $log(y)$ is the ...

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If $f: [a,b] \rightarrow \Bbb R$ is bounded and $c \in \Bbb R_+$. Show that for any partition $\{x_o,...,x_n\}$ of $[a,b]$ that $U(P,cf)= cU(P,f)$.
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2 votes

A stronger claim can be proven if $f$ has a Riemann integral, yet the process for the proof is the same. Suppose you have two functions $f(x)$ and $g(x)$ such that $g(x)=f(x/c)$ for some constant $c$,...

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Prove $\lvert\int_a^bf \cdot g\rvert \le (\int_a^bf^2)^{1/2} \cdot (\int_a^bg^2)^{1/2}$
2 votes

Consider the following integrals below: $$Q(x)=\int_{a}^{b}(xf(t)+g(t))^2dt$$ $$A:=\int_{a}^{b}f^2(t)dt, \ B:=2\int_{a}^{b}f(t)g(t)dt, \ C:=\int_{a}^{b}g^2(t)dt$$ Expressing $Q(x)$ in terms of $A$, ...

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Show that the sequence {$a_n$}, such that $a_1 =4$ and $a_{n+1}=3-{{2}\over\ a_n}$ is convergent to 2.
2 votes

To show it's monotone, consider the ratio, $$\frac {a_{n+1}}{a_n}=\frac{3-\frac 2{a_n}}{a_n}=\frac{3a_n-2}{a_n^2}<1$$ So, for $a_1=4$, the sequence $\left\{a_n\right\}_{n=1}^{\infty}$ is ...

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Interval of $x$ for a $7^{\text{th}}$ degree polynomial
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2 votes

Consider Taylor's Theorem, that is for $c\in (a,x)$: $$f(x)=\sum_{k=0}^{n-1}\frac {(x-a)^k}{k!}f^{(k)}(a)+\frac {(x-a)^n}{n!}f^{(n)}(c)$$ Note we call: $$R_n=\frac {(b-a)^n}{n!}f^{(n)}(c)$$ The ...

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Intuition of units: 6 chocolates $\div$ 2 friends = 6 $\times \frac 1 2$ units of what?
2 votes

First we must consider what this intuitively means. Let's say I have 10 chocolates, and 5 friends, and I want to distribute my chocolates evenly to each friend. Well, division provides us a clean and ...

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Confusion about radius of convergence of a power series
1 votes

You are indeed correct. For coefficients $a_k:=(1+\frac {1}{k})^{k^2}$, the Cauchy-Hadamard formula gives: $$R=\frac {1}{\limsup_{k\to\infty}(|a_k|)^{\frac {1}{k}}}=\frac {1}{\limsup_{k\to\infty}(|...

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Prove that $\exists\; \delta>0$ such that $\left\lvert \frac{f(t)-f(x)}{t-x}-f'(x)\right\rvert<\epsilon$
1 votes

Following from my hint I gave in the comments: $f'$ is continuous on the compact interval $[a,b]$, and thus is uniformly continuous on $[a,b]$. $\therefore \ \forall \epsilon>0$, $\exists \ \...

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Different values of $a$ in a linear system.
1 votes

$\mathbf{Hint:}$ I think you can, as stated in the comments and by the other answer, first triangularize/row reduce the given augmented matrix. For simplicity sake, suppose $[A \ | \ B]= \left[ \...

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Let $S=\{(x,y)\in\mathbb{R}^2:|x|\leq1 , |y|\leq1\}$ Prove $S$ is a closed set.
1 votes

We may first prove that if $U_1\subset\mathbb{R}^N$ and $U_2\subset \mathbb{R}^M$ are open sets, then $U_1\times U_2\subset \mathbb{R}^{N+M}$ is open. Suppose $U_1\subset \mathbb{R}^N$ and $U_2\...

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Limit of 4x + 1 as x approaches 0
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1 votes

We can employ a simple $\delta,\epsilon$ proof for $f(x)=3x+1$ as $x\to0$. Firstly, we can prove the conjecture proposed in part $\mathbb{b)}$: $\lim_{x\to0}f(x)=1$ That is, we want to show that $\...

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Triangle Inequality proof with three variables
1 votes

We can make use of two properties: $\mathbf{1}.$ For some constant $k\geq0$: $$|x|\leq k\iff-k\leq-x\leq k$$ $\mathbf{2.}$ For $x\in \mathbb{R}$, $$-|x|\leq x\leq |x|$$ The proof for the triangle ...

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Continuity and limits at end point of interval
1 votes

We say a function $f$ is continuous at some interior point $a$ $\iff$ $$\forall \epsilon >0, \exists \ \delta>0, \ s.t. \ |x-a|<\delta\implies|f(x)-f(a)|<\epsilon$$ Or more simply ...

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How to find intercepts of a function
1 votes

The equations are stated as follows: $$y=sinx \ \ \ \ \ \ \ \ y=sinx+0.5cos(2x)$$ The solution (point of intersection between both equations), can be determined via equality. Thus we have: $$sinx=...

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Calculating the value of a definite integral knowing the value of the integrand and its derivative on the boundaries.
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1 votes

Employ integration by parts. Note in general: $$\int_a^b fg'=\bigg[fg\bigg]_a^b-\int_a^b f'g$$ Thus, let $f=x$ and $g'=p''(x)$ $$\therefore \int_0^2 xp''(x)dx=\bigg[xp'(x)\bigg]_0^2-\int_0^2 p'(x)$...

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Sketching linear graphs
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1 votes

Plotting graphs is fundamental to mathematics, and is key to understanding the behaviour of polynomial, trigonometric, rational and exponential functions. This is the fundamental intuition behind ...

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Finding the cardinality of a set where $x$ is greater than $1$
1 votes

The cardinality is the amount of elements in a given set. For your set, $A = \left\{2x:x\in \mathbb{Z},-16\leq x\leq4\right\}$, $x$ takes on integer values between $-16$ and $4$. Note $2x$ is the ...

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Limit for some general variable
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1 votes

Let's first consider when $c=1$: We have the following: $$\lim_{t\to0}\frac {5ct+c-1}{5t^2+t}=\lim_{t\to 0}\frac{5t}{5t^2+t}=\lim_{t\to0}\frac{5t}{t(5t+1)}=\lim_{t\to0}\frac{5}{5t+1}=\frac 51=5$$ ...

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Analysis Integrable Function
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1 votes

The above function is a test for rationality. For $i=1,2,...,n$ and $n\in \mathbb{N}$, we define the following: $m_i=inf\left\{f(x):x\in [x_{i-1},x_i]\right\}$ $M_i=sup\left\{f(x):x\in[x_{i-1},x_i]\...

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Simplify expression, algebra
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1 votes

Note that $25^{n+2}-5^{2n+2}=5^{2n+2}(25-1)$ If this is at first not evident, consider the following: If you want to convert $25^{n+2}$ to some exponent of $5$, then: $$5^x=25^{n+2}\Rightarrow log(...

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