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. ...

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$)... View answer Accepted answer 4 votes For a sequence$\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$, $$m,n>N\Rightarrow |s_m-s_n|<\... View answer Accepted answer 4 votes Consider the power series of e^x, that is:$$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!}+\ ...$$Now consider Euler's constant e raised to ... View answer 4 votes We have the following:$$(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}{... View answer Accepted answer 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 ... View answer Accepted answer 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 ... View answer Accepted answer 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: $$\lim_{n\to\infty}a_n=L \ and \ \lim_{... View answer Accepted answer 2 votes We can directly evaluate each integral and solve for the unknown constant b. The following is given:$$\int_0^bxdx=\int_0^bx^2dx$$So:$$\int_0^bxdx=\bigg[\frac {x^2}{2}\bigg]^b_0=\frac {b^2}{2}$...

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

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 ...

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 $c$,...

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 $A$, ...

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 ...

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 ...

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 ...

You are indeed correct. For coefficients $a_k:=(1+\frac {1}{k})^{k^2}$, the Cauchy-Hadamard formula gives: $$R=\frac {1}{\limsup_{k\to\infty}(|a_k|)^{\frac {1}{k}}}=\frac {1}{\limsup_{k\to\infty}(|... View answer 1 votes Following from my hint I gave in the comments: f' is continuous on the compact interval [a,b], and thus is uniformly continuous on [a,b]. \therefore \ \forall \epsilon>0, \exists \ \... View answer 1 votes \mathbf{Hint:} I think you can, as stated in the comments and by the other answer, first triangularize/row reduce the given augmented matrix. For simplicity sake, suppose [A \ | \ B]= \left[ \... View answer 1 votes We may first prove that if U_1\subset\mathbb{R}^N and U_2\subset \mathbb{R}^M are open sets, then U_1\times U_2\subset \mathbb{R}^{N+M} is open. Suppose U_1\subset \mathbb{R}^N and U_2\... View answer Accepted answer 1 votes We can employ a simple \delta,\epsilon proof for f(x)=3x+1 as x\to0. Firstly, we can prove the conjecture proposed in part \mathbb{b)}: \lim_{x\to0}f(x)=1 That is, we want to show that \... View answer 1 votes We can make use of two properties: \mathbf{1}. For some constant k\geq0:$$|x|\leq k\iff-k\leq-x\leq k$$\mathbf{2.} For x\in \mathbb{R},$$-|x|\leq x\leq |x|$$The proof for the triangle ... View answer 1 votes We say a function f is continuous at some interior point a \iff$$\forall \epsilon >0, \exists \ \delta>0, \ s.t. \ |x-a|<\delta\implies|f(x)-f(a)|<\epsilon$$Or more simply ... View answer 1 votes The equations are stated as follows:$$y=sinx \ \ \ \ \ \ \ \ y=sinx+0.5cos(2x)$$The solution (point of intersection between both equations), can be determined via equality. Thus we have:$$sinx=...
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 $g'=p''(x)$ $$\therefore \int_0^2 xp''(x)dx=\bigg[xp'(x)\bigg]_0^2-\int_0^2 p'(x)... View answer Accepted answer 1 votes Plotting graphs is fundamental to mathematics, and is key to understanding the behaviour of polynomial, trigonometric, rational and exponential functions. This is the fundamental intuition behind ... View answer 1 votes The cardinality is the amount of elements in a given set. For your set, A = \left\{2x:x\in \mathbb{Z},-16\leq x\leq4\right\}, x takes on integer values between -16 and 4. Note 2x is the ... View answer Accepted answer 1 votes Let's first consider when c=1: We have the following:$$\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$$... View answer Accepted answer 1 votes The above function is a test for rationality. For i=1,2,...,n and n\in \mathbb{N}, we define the following: m_i=inf\left\{f(x):x\in [x_{i-1},x_i]\right\} M_i=sup\left\{f(x):x\in[x_{i-1},x_i]\... View answer Accepted answer 1 votes Note that 25^{n+2}-5^{2n+2}=5^{2n+2}(25-1) If this is at first not evident, consider the following: If you want to convert 25^{n+2} to some exponent of 5, then:$$5^x=25^{n+2}\Rightarrow log(...