All Questions
22
questions with no upvoted or accepted answers
6
votes
0
answers
535
views
Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$
Hi I am trying to integrate and obtain a closed form result for
$$
I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx.
$$
Here is what I tried (but I do not think this is ...
- 9,966
5
votes
0
answers
293
views
If $f$ is integrable, then $\sum\limits_{n\ge 1}\frac{1}{\sqrt n}\vert f(x-\sqrt n)\vert$ is almost everywhere finite
I would like to show that $$\sum_{n\ge0}\left\vert \frac{1}{\sqrt n} f \left(x-\sqrt n \right)\right\vert \tag{$*$}$$ converges for almost every (a.e.) $x$.
The only technique I have is based on the ...
- 3,918
2
votes
0
answers
365
views
A sum of two curious alternating binoharmonic series
Happy New Year 2024 Romania!
Here is a question proposed by Cornel Ioan Valean,
$$\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{2^{2n}}\binom{2n}{n}\sum_{k=1}^n (-1)^{k-1}\frac{H_k}{k}-\sum_{n=1}^{\infty}(-1)...
- 5,495
2
votes
0
answers
62
views
How to calculate limit of series using calculus
I have problem with calculating this limit, I know i should transform it into some integral but I don't know how:
$$
\lim _{ n\rightarrow \infty }{ \sum _{ k=0 }^{ n } (-1)^{ k }\binom nk\frac { f\...
- 461
2
votes
0
answers
68
views
calculating sum of a limit of integral
I am trying to calculate the following expression
$$
\sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy
$...
- 77
1
vote
0
answers
71
views
Evaluate $\int_1^e\frac{1}{x}dx$ using upper and lower sums
Using the upper and lower sums, I've tried to solve $\int_1^e\frac{1}{x}dx$ in the following way.
Let $$\Delta x = \frac{\mathit e -1}{n}$$ and let a partition $P$ be given by $$ P = \{1,1+\Delta x, 1+...
- 11
1
vote
0
answers
74
views
Finding the value of $ \sum_{n=1}^{\infty} \frac{2(2n+1)}{\exp( \frac{\pi(2n+1)}{2})+\exp ( \frac{3\pi(2n+1)}{2})}$
I have a question which askes to find the value of:
$$\displaystyle \tag*{} \sum \limits_{n=1}^{\infty} \dfrac{2(2n+1)}{\exp\left( \dfrac{\pi(2n+1)}{2}\right)+\exp \left( \dfrac{3\pi(2n+1)}{2}\right)}...
- 921
1
vote
0
answers
48
views
Value of a summation is greater than the corresponding integration
Let $$x[n] = 2^{-n}u[n + 1]$$ $$h[n] = 3^{n}u[-n+2]$$If we perform convolution, we get $$y[m]=x[n]*h[n] = \sum_{l=-\infty}^{+\infty}x[l]h[m-l] = \cases{\frac{27\times2^{3 - m}}{5} \ m \ge 2 \\ \frac{...
- 4,359
1
vote
0
answers
81
views
Prove that $1 < \sum\limits_{n = 1001}^{3001} \frac 1 n < \frac 3 2.$
Let $x = \sum\limits_{n = 1001}^{3001} \frac 1 n.$ Prove that $1 < x < \frac 3 2.$
My attempt $:$ What I did is as follows
By AM-HM inequality we have
\begin{align*} x =\sum\limits_{1001}^{3001}...
- 2,912
1
vote
0
answers
102
views
Searching for closed-form solutions to integral of dilogarithm
while trying to evaluate an infinite sum, I came across this integral:
$$ \displaystyle \mathcal{I}=\int_{0}^{\infty}\frac{\sin(x)}{x}\DeclareMathOperator{\dilog}{Li_{2}}\dilog\left(-e^{-x}\right)\...
1
vote
0
answers
175
views
Good upper bound for $\sum_{n=1}^N a^{-n} n^{-b}$, for $a, b \in (0, 1]$
Let $a, b \in (0, 1]$ and define $S_N:=\sum_{n=1}^N a^{-n} n^{-b}$.
Question
What is a good upper bound for $S_N$ ?
Observations
By a simple (and probably careless) application of Cauchy-Schwarz, ...
- 9,575
1
vote
0
answers
85
views
Interchange order of integrals and summation.
I would like to know, if we have:
$$I=\int_{0}^{a} (\ln y)\Big[\sum_{j=0}^{\infty}\frac{(-1)^j}{j!} y^{(j+1)b-1}\Big]\,dy,~~~~a,b>0$$
then interchange order of integral and summation,
$$I=\sum_{j=...
- 515
1
vote
0
answers
83
views
Analogue for finite sums of $\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g)$
Please help me to find an analogue for finite sums of
$$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) \tag{*}$$
where $ \bar{f}(t)=\frac{1}{t}\int_{a}^{t} ...
- 11
0
votes
0
answers
64
views
Converting complex-exponential summation to Fresnel integrals
I have a summation
$$S = \sum_{n=0}^N e^{-jn^2a}, \ a\ne 0, \ n\in\{0,1,\cdots,N\}$$
and it can be approximated by
$$S\approx I = \int_{n=0}^N e^{-jn^2a}dn$$
when $N$ is sufficiently large. ...
- 71
0
votes
0
answers
46
views
Why can we approximate a sum by a definite integral?
From wikipedia https://en.wikipedia.org/wiki/Summation#Approximation_by_definite_integrals, I read that
$\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds$ for increasing ...
- 365