Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 199

Show that for any $n$, $sum_{k=0}^infty frac{1}{3^k}binom{n+k log(3)/log(2)}{k}=c 2^n$ for some constant $c$

I stumbled upon this from playing around and it seems to be true but I don't know how to approach it....

Asked on 10/15/2020

2 answer

If $frac{a}{b}$ is irreducible, then the quotient of the product of any $2$ factors of $a$ and any $2$ factors of $b$ are irreducible.

$a,bin mathbb{Z}$ Factors of $a$: $a_1,a_2,...,a_n$. Factors of $b$: $b_1,b_2,...,b_m$Prove that if $frac{a}{b}$ is irreducible, then$frac{a_ia_j}{b_kb_l}$ is irreducible for all ...

Asked on 10/15/2020 by Simplex1

3 answer

cell structure of $S^2times S^2$ with $S^2times {p}$ identified to a point

I'm preparing for an exam and found this problem: Let $X$ be obtained from $S^2times S^2$ by identifying $S^2times {p}$ to a point, then what is ...

Asked on 10/15/2020 by user830531

1 answer

How does strong convexity behave under Minkowski sums?

Suppose we have two convex sets $A,B$. It is straightforward to show the sum $A+B = {a+b:a in A,bin B}$ is convex. For take any $c=a+b$ and...

Asked on 10/14/2020 by Daron

1 answer

Miscellaneous Problem Chapter I ex.22 on G.H.Hardy's book "A course of pure mathematics"

If all the values of x and y given by$$ax^{2}+2hxy+by^{2}=1, a'x^{2}+2h'xy+b'y^2=1$$(where $a, h, b, a', h', b'$ are rational) are rational, then$$...

Asked on 10/14/2020 by SAUL ROJAS

0 answer

Must a series converge after a finite number of 'césaro mean' applications if it does after infinitely many.

Let $alpha=(alpha_n)_{nge0}$ be a sequence of real numbers. Let $C(alpha)=(beta_n)_{nge0}$ where$$beta_n=frac{1}{n+1}sum_{k=0}^{n}alpha_k$$Finally let $C^n=Ccirc C^{n-1}$ with $C^0=text{Id}$ be functions from the set of real...

Asked on 10/14/2020 by Oddly asymmetric

0 answer

Why am I getting derivative of $y = 1/x$ function as $0$?

I was finding the derivative of the function: $y = 1/x$. I did the followed steps: begin{align*}frac{frac{1}{x+dx} - frac{1}{x}}{dx} &=left(frac{1}{x+dx} - frac{1}{x} right) frac{1}{dx} \&= frac{1}{x...

Asked on 10/13/2020 by Dhruv Agarwal

4 answer

How to determine the span of two vectors: $(4,2)$ and $(1, 3)$

How to determine the span of two vectors in $mathbb R^2$: $(4,2)$ and $(1, 3)$ Do I subtract them? I don't how I'd solve this. Thanks in...

Asked on 10/13/2020 by Doctor Reality

4 answer

Putnam 2018 - Exercise A.5 - proof check

The problem statement is as follows. Let $f:Bbb R to Bbb R$ be an infinitely differentiable function satisfying $f(0) = 0$ and $f(1) = 1$, and ...

Asked on 10/11/2020 by dfnu

0 answer

Difference of Consecutive Terms in a Recurrence Sequence

I have a question that seems simple, but it has caused me some trouble when trying to prove it. Given a recurrence relation with non-negative integer coefficients,$$G_{n+1}...

Asked on 10/10/2020 by ThomasMart

1 answer

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