Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 118

Proof that $ frac{2^{-x}-1}{x} = sum_{n=0}^{infty} frac{ (-1)^{n+1}x^n(ln2)^{n+1}}{(n+1)!} $

If$$2^x = sum_{n=0}^{infty} frac{(xln(2))^n}{n!} hspace{1cm} forall x in mathbb{R},$$Proof that:$$ frac{2^{-x}-1}{x} = sum_{n=0}^{infty} frac{ (-1)^{n+1}x^n(ln2)^{n+1}}{(n+1)!} $$I did the following: begin{align*}&2^x =...

Asked on 02/14/2021 by Sugaku

3 answer

$24ml+1=k^2$ has no solution for all $l=1 dots m$

Investigating solutions of $$24ml+1=k^2$$ for $l=1dots m$ The question is to find the $m$-s for which the above equation has no solution for all $l=1..m$-s.The...

Asked on 02/14/2021 by Gevorg Hmayakyan

1 answer

I can't find the analytical expression of the sum from this probability problem

It originates from this problem:We have a box with $n$ balls that are evenly mixed, all are$color{blue}{blue}$ except for one that is $color{red}{red}$.If we pick blindly...

Asked on 02/14/2021

1 answer

Slightly confused about the commutator subgroup

Let's look at the group of all permutations of a set $left{ 1,2,3 right}$. There are exactly 6 elements in this group: $left( begin{matrix} 1...

Asked on 02/14/2021

1 answer

Isomorphism from ring to ring. Exercise from General Topology (J. Kelley)

I tried Exercise K from chapter 2 of General Topology by Kelley. Is this enough to show isomorphism between Boolean rings? This is part (d).Let $X$ be a set...

Asked on 02/14/2021 by flowian

1 answer

Polynomial with root $α = sqrt{2}+sqrt{5}$ and using it to simplify $α^6$

Find a polynomial $space P(X) in mathbb{Q}[X]space$ of degree 4 such that $$alpha = sqrt{2} + sqrt{5}$$ Is a root of $P$. Using this polynomial, find...

Asked on 02/14/2021

4 answer

$sigma(n)$ is injective?

I was reading about the sum function of divisors $sigma(n)$, and the question arose as to whether this is injective. Does anyone know the answer? in case of being...

Asked on 02/14/2021 by ferphi

1 answer

Polynomial multiplication is associative

Let $A$ be a ring and $f,g,hin A[X]$. I want to show that $(fg)h=f(gh)$, where $(xy)_n:=sum_{n=j+k}x_jy_k$ for all $ninmathbb{N}$. Attempt: Let $ninmathbb{N}$. I have...

Asked on 02/14/2021

0 answer

Find area of equilateral $Delta ABC $

I'm going to solve this geometry problem without using rotational , any other way like coordinates : Problem : Given an equilateral triangle $Delta ABC $ such that :...

Asked on 02/14/2021 by Ellen Ellen

2 answer

Convergence in distribution of $N(0, 1/n)$

I am a graduate student in statistics and am self-studying convergence in probability. I am a little confused on the following problem. I am trying to follow a proof...

Asked on 02/13/2021

1 answer

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