I want to use absolute value notation to represent the phrase "$x$ is within 3 units of $c$". Which of the following is the correct answer? (i) or (ii)? (i) $|x-c|<3$, (ii) $|x-c|\leq{3}$
Prove the Riemann Sum to the $n$th term with $i=1$ of $ar^{i-1}$ is equivalent to $a\frac{r^n -1}{r-1}$. This is true for a geometric series when $r$ doesn't equal $1$.
Let $N_t$ a renewal process with $T_i$ the jumps. I know $U(t)=E(N_t)$ on [0,t]. Let $B_t=T_{N_t}-t$ and $G_t(u)=P(B_t \leq u)$How to show on ]t,t+a]: $E(N_{t+a}-N_t)=\int_0^a U(a-u)G_t(du)=(G_t \star U)(a)=(U \star G_t)(a)$Thank you
Consider the set of all functions on [0,1] of the form $h(x) = \Sigma_{j=1}^na_je^{b_jx},$ where $a_j, b_j \in \mathbb{R}$. Is this set closed under addition? That is consider $f(x), g(x) \in \mathcal{B}$, where $\mathcal{B}=\{\text{set of functions
What is the precise mathematical meaning of the phrase "for all $x$ close to $c$"? Does it mean that for all $x$ that satisfy the inequality $|x-c|<\epsilon$, where $\epsilon$ is a small positive number? This phrase is used very often.
I trying to find the value for which $n^2 -n +1$ is less than $ 6n\log_{2}{n} +2n $ where n is a power of $2$.Trying it iteratively using a CAS you find that $n = 64$.How can $n^2 -n +1 < 6n\log_{2}{n} +2n $ where n is a power of $2$ be solved analyt
Let $x_{1},x_{2}, \ldots , x_{k}$ be positive real numbers such that$$x_{1}+ x_{2}+ \cdots +x_{k}\geq1$$ $$0\leq x_{i}\leq1\text{ for }i \in \{1,2,\ldots ,k\}$$I want to prove the following inequality $$(1-x_{1})(1-x_{2})\cdots(1-x_{k}) \leq \left(1-
Let $f:[0,1]\to [0,1]$ be a function such that$f(1)=1$ $f(x)+f(y)\leq f(x+y)$, for any numbers $x$ , $y$ , $x+y \in [0,1]$ Then we have to show that $f(x)\leq x$ for any $x\in [0,1]$.I can see that $f(0)=0$, and am able to get a few more properties b
Find the values of $x$ for which function is defined: $f(x) = \log(5x^x-4)+\sqrt{x-1}$.$ \log(5x^x-4) > 0 \Rightarrow 5x^x-4 > 1 \Rightarrow 5x^x-5 > 0 $ $x = \frac{ 8 \pm \sqrt{160}}{10}$ also $x-1 > 0 \Rightarrow x >1$Please g
In dealing with inequalities I've run into a certain peculiarity which I am currently unable to explain.The example: Find the interval of time during which the ball is at least 32 feet above ground.h = -16t^2 + 16t + 128 // Height of the ball in feet
Which of the following have the least value if $-1 < x < 0$?(A) $-x$(B) $1/x$(C)$-1/x$(D)$1/x^2 $(E)$1/x^3$ I'm not sure what to do, but I'll definitely try. We can break it up into two inequalities: $ x > -1$ and $0 > x$$\implies -x < 1, 0
Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_h$ and $K_f$ are rates of return on home and foreign currency if it is invested over a period $T$. (A) Assume that the forward rate of exchange $F$ satisfies
For the ring $\mathbb{Z}[\sqrt{d}]$, we define the norm function $f:\mathbb{Z}[\sqrt {d}] \rightarrow \mathbb{N}\cup \{0\}$ by: $ f(a+b\sqrt d)=|a^2-db^2|$. How can I prove that $f(x)=0$ if and only if $x=0$?What I've got: Suppose $x=a+b\sqrt d$ and
the question was: What is the infimum from $K=\{x ~ |~0 \leq x \lt 1\}$ for the partial order $\{ (x,y) ~ | ~ ( \exists z \in \mathbb{R}_{\geq 0})[x-z=y] \}$ on set $\mathbb{R}$. This was a question in a math quiz from our prof and the answer was $1$
I have recently learnt the comparison test, MCT, Fatou's Lemma and DCT for Lebesgue integrals, but have been struggling with the details of the proofs.1) f is 0 a.e. so is integrable to 02) ...
Evaluate $$\int_{-\infty}^\infty xe^{-m(x-a)^2}$$ where $m$ and $a$ are constants I can solve this if the exponential is simple $e^{x^2}$ by substitution, but this one doesn't work that way as when I subsitute for $x-a$ I run into problems and often
How many numbers between $1$ and $9999$ have sum of their digits equal to $8$? $16$? Can someone tell me if I got the right answers? I solved both cases and I've got $148$ for $8$ and $633$ for $16$. I solved this problem using $x_1+x_2+x_3+x_4=8$ th