Sequences and Series

Let $\xi_n$ be a Poisson random variable with $\lambda = n \in \mathbb{N}$. That is $P(\xi_n = k) = \frac{n^ke^{-n}}{k!}$, for $k \in \mathbb{N}_0$. Let $f_+(n) = P(\xi_n \geq n),\ f_-(n) = P(\xi_n &g...

Prove that any unbounded sequence has a subsequence that diverges to $\infty$.

I know the solution is 1/e but i don't know how to get there. 

Let $\gamma >0$ and $r=(r_n)_{n \in \mathbb{Z} } \in \ell^\infty(\mathbb{Z})$ (a bounded and real-valued sequence), where $r_n \geq 0 \ \forall \ n \in \mathbb{Z}$. Consider the function $f(x)=\sum...

Question is provided in the picture

A) Find the Taylor Polynomials of orders 0, 1, 2, 3 generated by $f(x) = sinx$  at $a = \frac{\pi}{4} $ .B) Find the Mclaurin series for $f(x) = e^{\frac{x}{2}} $ .

In my Calculus class, we were solving an integration by parts problem two different ways. One way gave an answer pretty easily, the other led to an infinite series that stumped us. After playing aroun...

Please read the following scenario, modify the given mathematical model as instructed, and present your model and analysis. Please respond using difference equations (not differential equations).Non-L...

Determine if the following series converge absolutely, conditionally, or diverge

See the attachment below.If you're unable to solve it try to find a research paper.

$\sum_{k=1}^{n} \tfrac{2k+1}{k^{2}(k+1)^2 } $This is for a discrete math course. Lots of students were strugglign with this one, so the instructor gave us a couple hints: first, we are supposed to use...

Below is a sequence I would like to convert into its mathematical representation: [1 01 10 010 11 011 101 0101 110 0110 1010 01010 111 0111 1011 01011 1101 01101 10101 010101 1110 01110 10110 010110 1...

tan(φ)=φ+(1/3)φ^3wrapping a cable around the earth once then adding one metre to the cable (radius is R=6378km)Find two expressions for the length of the cable. Use these together with your Maclaurin...

If you noticed, without the summation notation, it is the present value money formula with finite time period. PV - present value FV - future value i - interest rate n - number of yearsPV = FV/(1+i)^n...

Use induction to prove that $1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}} \leq 2 \sqrt{n}-1,$
for all $n\geq 1$.

Show that $\sum_{n=1}^{\infty} \frac{\sin n}{n}$ is convergent. I was trying to use the Dirichlet test, but couldn't verify that the conditions of the Dirichlet test are satisfied. 

Using this iteration method, I need to figure out the closed form for this reccurance relation:T(n) = 2T(n - 10) + 3, for n > 11, T(n) = 5 for n <= 10The answer is to be expressed exactly, using...

Hello,I need this done asap. Questions 1-10.I ask for it to be comepleted using methods that are for the most part easy to understand so i can use it to study for the future.im not really sure if this...

Hello,I am asking you to do all the questions on the attached doucments to completion. Please provide full answers, and please solve them in a way that a someone who semi-understands (me) can comprehe...

Is the infinite series \sum_{i=1}^{\infty}\arctan (\frac{n+1}{n^2+5}) convergent or divergent?

Below 

Pls find attached 

Question from a problem set. Definitions given:Convergent Sequence: A sequence $(x_n)$ in a metric space $X$ converges to $x_0$ if for every $\varepsilon>0$ there is $n_0\in\mathbb{N}$ such that $d...

Is $\sum_{n=1}^{\infty}\frac{\arctan (n!)}{n^2}$ convergent or divergent?Which test should I use? I feel so dumb!