## Ratio test for $\frac{\sqrt{n^n}}{2^n}$ February 11    2

Before I tried root test, I did ratio test for $\frac{\sqrt{n^n}}{2^n}$ and got: $$\lim_{n\to\infty}\frac{\sqrt{(n+1)^{(n+1)}}}{2^{n+1}}\cdot \frac{2^n}{\sqrt{n^n}} = \lim_{n\to\infty} \frac{1}{2}\cdot \sqrt{\frac{(n+1)^{(n+1)}}{n^n}} = \frac{1}{2}$$

## Ratio test: $n^\sqrt{n}$ February 11    4

I need to determine the radius of convergence of: $$\sum_{n=1}^\infty z^n n^\sqrt{n}$$I have, by use of the ratio test, written: (Because I know it tends to 1) $$\lim_{n\to\infty} \frac{n^\sqrt{n}}{(n+1)^\sqrt{n+1}}\to 1$$But I am unable to demonstra

## Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$ February 11

As a consequence of this Q, I need some help evaluating the following integral: $$\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$$Integration by parts wouldn't simplify things and I guess that a subtle manipulation on the integrand is needed.

## What about $\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n$ when $y=x\to\infty$ February 11

For a real $x\geq 2$ and when we take $y= [x]$ its integer part, I am trying to study the asymptotic size or growth of $$\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n,$$ I believe that combining Abel's summation formula with known theorem

## Independence, Inverse and Additive Identity for vector with defined vector addition and scalar multiplication February 11    1

I am struggling to find my bearings on this question. I am confident that I can do parts a and d. I have no clue how to approach b. I was also wondering if the redefined vector addition and scalar multiplication will effect part c?If you could help m

I need to prove that the vector space of $\mathbb{R}^2$ with the following operations:$x + y = (x_1 + 2y_1, 3x_2 - y_2)$The usual scalar multiplication of $cx = (cx_1, cx_2)$ The answers in my book say that axiom $\textbf{4}$ fails to hold that is th

## Conditional expectation of a random walk given that it is positive February 11

Let $\{\xi_k\}$ is a sequence of iid random variables with $E(\xi_1)=0$ and $E(\xi_1)^2=\sigma^2<\infty$. Define the random walk $Y_n=\sum_{k=1}^n \xi_k$. Is it necessarily true that the conditional expectation $E(Y_n|Y_n>0)=\mathcal{O}(\sqrt{n})$ a

## Given that 6 men and 6 women are divided into pairs, what is the probability that none of the women will sit with a man February 11

I've generalized the question I was given here for simplicity: 6 men and 6 women are to be paired for a bus trip. If the pairings are done randomly, what's the probability that no women will end up sitting next to a man? Here's my first attempt, but

## Why is the sequence $u_N = \inf\{s_n : n \gt N\}$ increasing February 11

A question in my book I am studying says to let $s_n$ and $t_n$ be sequences and suppose there exists $N_0$ such that $s_n \le t_n$ for all $n \gt N_0$. Show $\lim \inf s_n \le \lim \inf t_n$ and $\lim \sup s_n \le \lim \sup t_n$.The hint for the pro

## Inference Rules, Not(P) Implies Not(Q) / Q Implies P February 11    1

I do not understand Implication and Inference, I am going over the MIT Computer Science course and they have this part in their lecture notes, why is the second rule not a logical deduction? ...

## Do path homotopy classes of concatenated paths have a middle fixed point February 11    1

If $[a]$ and $[b]$ are path homotopy classes, then $[a].[b]$ is defined as $[a*b]$, where $a*b$ is defined as the concatenation of the paths $a$ and $b$. Let us say $a(1)=b(0)=p$. Then does each path in $[a*b]$ have to contain p? No they don't have t

## Prime Counting: In truncation rule #2 mentioned in an AMS.org article, I'm unclear how special leaves work February 11

I'm reading through an AMS.org article on prime counting. The article covers the history of prime counting and focuses on improvements to the Meissel-Lehmer method. It is the improvement on page 236 which I am having trouble following.First, the arti

## SQL joins and analysis February 11

Say we have a users table and an events table and what sort of analysis can be done? Also, what is some SQL statements to describe the analysis of these 2 tables?

## $2^k+3$ : Brute Forcing Theory Below The Square Root February 11

I'm testing a theory of brute forcing $2^k+3$.I've tried to test $(2^k)+3$ where $k=84$ but my computer just takes too long... Java takes too long too..It's pretty stupid to assume 83 tests makes a perfect case, but I'll take it as a halfway decent c

## Simplifying Expression Factorial Expression February 11    2

I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm trying to figure out how I simplify.ThanksAs noted in the

## Parametrising a curve using curvature and torsion functions February 11

I am trying to get a parametrization of the curve whose curvature and torsion functions are given as $$\kappa(s)= \dfrac{1}{1+s^2} ,\;\; \tau(s) = \dfrac{s}{1+s^2}$$I know that in general it is not possible to get parametrizations from the curvature

## Proof: For every positive integer $n$, there is a sequence of $n$ consecutive positive integers containing no primes. February 11

Let $x=(n+1)!+2$. I get how to prove that $x$ or $x+1$ is prime, but there is a step in my book that proves that $x+i$ is prime like this:$x+i=(1)(2)(3)(4)....(n+1)+(i+2)$. But then it factors out to this: $x+i=(1)(2)(3)(i+1)(i+3)...(n+1)+1$.How does

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