determinant of the matrix with power series element

by jokerjoestar, Apr 3, 2025, 4:31 PM

Given the function

$f_k(x) = 1 + 2x + 3x^2 + \dots + (k+1)x^k,$
show that

$\begin{vmatrix} f_0(1) & f_1(1) & f_2(1) & \dots & f_{2023}(1) \\ f_0(2) & f_1(2) & f_2(2) & \dots & f_{2023}(2) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ f_0(2024) & f_1(2024) & f_2(2024) & \dots & f_{2023}(2024) \end{vmatrix} = \prod_{k=1}^{2024} k!.$

linear algebra

matrix

function

A hard inequality

by Butterfly, Apr 3, 2025, 9:43 AM

let $x_1,x_2,\cdots$ be all of the extrem points of $f(x)=x\cos\left(\frac{1}{x}\right)(x>0)$ . Prove $f^2(x_1)+f^2(x_2)+\cdots \le \frac{\pi}{6{\rm e}}.$

inequalities

calculus

An interesting limit

by Alphaamss, Apr 3, 2025, 9:16 AM

Suppose $x_1=\frac\pi2,\quad x_{n+1}=x_n-\frac{\sin x_n}{n+1},$ I can prove that the sequence $\{nx_n\}$ is convergent by monotone bounded convergence theorem.
Is there any method to compute the limit of $\{nx_n\}$ , or give the asymptotic representation of $\{nx_n\}$ ? Any help and hints will welcome!

limit

calculus

Real & Imaginary Parts

by Entrepreneur, Apr 3, 2025, 8:51 AM

Evaluate $\Re\left(\frac{\Gamma(ix)}{\Gamma(ix+\frac 12)}\right)\;\&\;\Im\left(\frac{\Gamma(ix)}{\Gamma(ix+\frac 12)}\right).$

calculus

complex numbers

Proving AB-BA is singular from given conditions

by Ciobi_, Apr 2, 2025, 2:04 PM

Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$ . Prove that:
a) if $n$ is odd, then $\det(AB-BA)=0$ ;
b) if $\text{tr}(A)\neq \text{tr}(B)$ , then $\det(AB-BA)=0$ .

linear algebra

trace

matrix

Easy matrix equation involving invertibility

by Ciobi_, Apr 2, 2025, 1:46 PM

Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$ , for any $k \in \{1,2,\dots ,n\}$ . The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$ . Prove that $A$ and $B$ are invertible.

Strange limit

by Snoop76, Mar 29, 2025, 7:42 AM

Find: $\lim_{n \to \infty} n\cdot\sum_{k=1}^n \frac 1 {k(n-k)!}$

limit

Summation

real analysis

Matrix problem

by hef4875, Mar 26, 2025, 9:49 AM

The matrix $A = (a_{ij}) \in Mat_p(\mathbb{C})$ is defined by the conditions
$a_{12} = a_{23} = \dots = a_{(p-1)p} = 1$ and $a_{ij} = 0$ for a set of indices $(i,j)$ .
Prove that there do not exist nonzero matrices $B, C \in Mat_p(\mathbb{C})$ satisfying the equation
$(I_p + A)^n = B^n + C^n.$ $\forall$ $n$ is a postive integer.

This post has been edited 2 times. Last edited by hef4875, Mar 26, 2025, 9:51 AM

linear algebra

matrix

Binomial inequality

by Snoop76, Feb 2, 2025, 5:08 PM

Is it true? $\sum_{k=0}^n (2k-1)!!{n\choose k} >\left(\frac{2n}{e}\right)^n\sqrt{2e}$

combinatorics

inequalities

(10)
(+)

Putnam 2008 A5

by Kent Merryfield, Dec 8, 2008, 12:23 AM

Let $n\ge 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials with real coefficients such that the points $(f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $\mathbb{R}^2$ are the vertices of a regular $n$ -gon in counterclockwise order. Prove that at least one of $f(x)$ and $g(x)$ has degree greater than or equal to $n-1.$

(32)
(+)

Submit

thank you both wahoo
by Helena_Liang, Feb 16, 2025, 3:45 AM
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by Helena_Liang, Feb 15, 2025, 4:50 AM
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thank you sir but you should admit orz
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by alice_inmathland, Jan 22, 2025, 7:49 PM
pls pls be active again :eyes:
by Hestu_the_Bestu, Nov 16, 2024, 12:45 AM
I have been inactive on aops
perhaps I'll be active again
by Helena_Liang, Oct 9, 2024, 3:10 AM
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by mathical8, Sep 29, 2024, 8:12 PM
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by Hestu_the_Bestu, Sep 9, 2024, 9:15 PM
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also u should post something
by alice_inmathland, Sep 7, 2024, 11:36 PM
contrib

1434 orz xooks
by ChromeRaptor777, Aug 30, 2024, 4:04 AM
@2 below ay yes sir I shall
@below indeed
by Helena_Liang, Aug 22, 2024, 1:34 AM
first as well $\text{[math]}$
by madeleinelee, Nov 21, 2023, 8:26 PM
sapphire's css skills are pro
by Helena_Liang, Nov 20, 2023, 5:14 PM
Unfortunately, first
by awesomeming327., Nov 20, 2023, 4:59 AM
Lol Helena and Emma I think your blogs are essentially the same as mine ha cant top that
by SapphireDolphin9, Nov 20, 2023, 4:28 AM
first $\text{[math]}$
by awesomeming327., Nov 20, 2023, 3:54 AM
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by flec, Nov 20, 2023, 3:35 AM
I agree with that statement
this blog would not be complete without flec
by Helena_Liang, Nov 20, 2023, 3:34 AM
i had very valuable edits
by flec, Nov 20, 2023, 1:38 AM
lol emma I think your blog css is essentially the same as ours (with edits)
by Helena_Liang, Nov 19, 2023, 11:46 PM
fifth!! also very nice css and blog idea :D $~$
by Beast_Academic_Emma, Nov 19, 2023, 10:28 PM
fourth!!
by mathical8, Nov 19, 2023, 6:49 PM
third ! $\text{[math]}$
by purplepenguin2, Nov 19, 2023, 1:41 AM
second : $\text{[math]}$ D
by Helena_Liang, Nov 18, 2023, 6:26 AM
obviously i take the first shout
by flec, Nov 18, 2023, 6:07 AM

70 shouts

two up, one down

by jokerjoestar, Apr 3, 2025, 4:31 PM

by Butterfly, Apr 3, 2025, 9:43 AM

by Alphaamss, Apr 3, 2025, 9:16 AM

by Entrepreneur, Apr 3, 2025, 8:51 AM

by Ciobi_, Apr 2, 2025, 2:04 PM

by Ciobi_, Apr 2, 2025, 1:46 PM

by Snoop76, Mar 29, 2025, 7:42 AM

by hef4875, Mar 26, 2025, 9:49 AM

by Snoop76, Feb 2, 2025, 5:08 PM

by Kent Merryfield, Dec 8, 2008, 12:23 AM