All Questions
25
questions
3
votes
1
answer
119
views
Does $\prod\limits_{n=1}^{\infty}\int\limits_{-1}^{1}\left|\frac{\lfloor t^{1-n}\rfloor}{\lfloor t^{-n}\rfloor}\right|dt$ converge to $1$?
This is the first $40$ partial products on Desmos (Desmos gave me "undefined" for everything higher):
Looking at this, it doesn't appear to converge but then, while attempting to get Desmos ...
- 1,688
5
votes
1
answer
179
views
What is $\int\limits_0^1\left\{\frac{1}{x\left\{\frac{1}{x}\right\}}\right\}dx$?
$\{x\}$ is the fractional part of $x$.
$\{x\}=x-\lfloor x\rfloor$
I ended up with this double summation:
$$\lim\limits_{\substack{a\to\infty\\b\to\infty}}\sum_{m=1}^{a}\sum_{n=0}^{b}\left(\frac{1}{m}\...
- 1,688
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
4
votes
1
answer
130
views
Solving the integral $\int\limits^{\infty}_{1}\frac{x-\sqrt{\lfloor x^2\rfloor}}{x}dx$
This problem was just for fun
Here was what I managed to come up with:
The integral is approximately equal to $0.242070053984$.
The integral equals
$$\sum_{n=1}^{\infty}\int_{\sqrt{n}}^{\sqrt{n+1}}...
- 1,688
0
votes
0
answers
46
views
Why can we approximate a sum by a definite integral?
From wikipedia https://en.wikipedia.org/wiki/Summation#Approximation_by_definite_integrals, I read that
$\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds$ for increasing ...
- 365
4
votes
2
answers
202
views
Evaluate the following humongous expression
PROBLEM:
Evaluate $$\left(\frac{\displaystyle\sum_{n=-\infty}^{\infty}\frac{1}{1+n^2}}{\operatorname{coth}(\pi)}\right)^2$$
CONTEXT:
I saw a very interesting and yet intimidating question on the ...
- 1,253
1
vote
1
answer
56
views
Calculating bounds for a certain expression using Faulhaber
Let $F(n,k)=\sum_{i=1}^ni^k$ for $k\geq 0$ and $n\geq 1$. I would like to prove
$$\sum_{i=1}^{n-1}i^k\leq\frac{n^{k+1}}{k+1}\leq\sum_{i=1}^ni^k.$$
The idea is to use integration where the upper and ...
- 43
0
votes
0
answers
95
views
Proof of $sin$ formula.
I am reading this quesiton and accepted answer.
Question is about proof.
$S = \sin{(a)} + \sin{(a+d)} + \cdots + \sin{(a+nd)}$
$S \times \sin\Bigl(\frac{d}{2}\Bigr) = \sin{(a)}\sin\Bigl(\frac{d}{2}\...
- 879
9
votes
6
answers
376
views
Evaluate : $S=\frac{1}{1\cdot2\cdot3}+\frac{1}{5\cdot6\cdot7}+\frac{1}{9\cdot10\cdot11}+\cdots$
Evaluate:$$S=\frac{1}{1\cdot2\cdot3}+\frac{1}{5\cdot6\cdot7} + \frac{1}{9\cdot10\cdot11}+\cdots$$to infinite terms
My Attempt:
The given series$$S=\sum_{i=0}^\infty \frac{1}{(4i+1)(4i+2)(4i+3)} =\sum_{...
- 9,599
1
vote
1
answer
105
views
Why is $\sum a_nf(n) = \int_0^xf(t)~d(A(t))$?
This equation is a part of Abel's summation formula
$a_n$ is a sequence, $f$ is a real differentiable function such that $f'$ is Riemann integrable.$$A(x) = \sum_{1\le n\le x}a_n$$
I don't see why is $...
- 1,641
1
vote
2
answers
100
views
Finding a closed form expression of a sequence that is defined recursively via a definite integral
Consider the following series function that is defined recursively by the following definite integral
$$
f_n(x) = \int_0^x u^n f_{n-1}(u) \, \mathrm{d}u \qquad\qquad (n \ge 1) \, ,
$$
with $f_0 (x) = ...
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:
$$ \displaystyle \mathcal{I}=\int_{0}^{\infty}\frac{\sin(x)}{x}\DeclareMathOperator{\dilog}{Li_{2}}\dilog\left(-e^{-x}\right)\...
6
votes
2
answers
642
views
Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$
How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
- 1,132
0
votes
1
answer
96
views
How to show that $\int_0^1x^{-x}dx = \sum_{n=1}^\infty n^{-n}$? [duplicate]
How would I go about showing that $\int_0^1x^{-x}dx = \sum_{n=1}^\infty n^{-n}$
Right now my numerical analysis class is covering gaussian quadrature but we have also covered interpolation. I'm not ...
0
votes
1
answer
39
views
Solving integrals with power series
Okay, so I'm looking at the anwers to a question where you're supposed to solve a definite integral depending on $x$. And I do not understand the equality below:
$$\int_{0}^{1} \frac{t^2}{1-tx} dt = \...
- 41