Double derivative coated integral

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0 replies

jlacosta

Apr 2, 2025

0 replies

Sum of Series

P162008 0

an hour ago

Let $S_n = \sum_{r=1}^{n} r.2^{\frac{r}{2}(1 + (-1)^r)}. 3^{\frac{r}{2}( 1 - (-1)^r)}.$ If $S_{20} = \alpha.3^{21} + \beta.2^{22} + \frac{391}{288}$ then find the value of $\left[\frac{32\alpha - 9\beta}{4}\right]$ where $\left[.\right]$ denotes the greatest integer function.

0 replies

P162008

an hour ago

0 replies

Triple Sum

P162008 0

an hour ago

Evaluate $\Omega = \sum_{k=1}^{\infty} \sum_{n=k}^{\infty} \sum_{m=1}^{n} \frac{1}{n(n+1)(n+2)km^2}$

0 replies

P162008

an hour ago

0 replies

Infinite Limit

P162008 0

an hour ago

Find the value of limit
$\Omega = \prod_{n=2}^{\infty} \frac{n^{14} - n^{12} - 2n^8 + 2n^6 + n^2 - 1}{n^{14}}$

0 replies

P162008

an hour ago

0 replies

3-dimensional matrix system

loup blanc 2

N an hour ago by loup blanc

Let $A=\begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}$ .
i) Find the matrices $B\in M_3(\mathbb{R})$ s.t. $A^TA=B^TB,AA^T=BB^T$ .
EDIT. ii) Show that each solution of i) is in $M_3(K)$ , where $K=\mathbb{Q}[\cos(\dfrac{2}{3}\arctan(3\sqrt{3}))]$ .
iii) Solve i) when $B\in M_3(\mathbb{C})$ .
EDIT. In iii) we again consider the transpose of $B$ and not its conjugate transpose.

2 replies

loup blanc

Mar 15, 2025

loup blanc

an hour ago

No more topics!

Double derivative coated integral

RenheMiResembleRice 1

N Mar 28, 2025 by HacheB2031

Source: Lufang Yue, Lianru Meng

Show that J>0. For this to be true, I guess we also need the condition for c and d to be positive, isn't that right?

1 reply

RenheMiResembleRice

Mar 28, 2025

HacheB2031

Mar 28, 2025

Double derivative coated integral

G H J

Reveal topic tags

calculus

derivative

integration

G H BBookmark kLocked kLocked NReply

Source: Lufang Yue, Lianru Meng

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RenheMiResembleRice

259 posts

RenheMiResembleRice

#1 h Mar 28, 2025, 3:13 AM

Y by

Show that J>0. For this to be true, I guess we also need the condition for c and d to be positive, isn't that right?

Attachments:

This post has been edited 1 time. Last edited by RenheMiResembleRice, Mar 28, 2025, 3:14 AM

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HacheB2031

386 posts

HacheB2031

#2 h Mar 28, 2025, 5:07 AM

Y by

there's infinite functions where $J<0$
like $H(x)=-e^x,$ $H'(x)=-e^x,$ $\displaystyle J=\int_c^dH''(x)\,\text dx=H'(d)-H'(c)=e^c-e^d,$ and since $c<d$ then $e^c<e^d$ and so $e^c-e^d=J<0$
there would need to be some restriction on $H$

Z K Y

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