All Questions
239
questions
2
votes
1
answer
177
views
Show that $ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)} $
Show that $$ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)}
$$
My attempt
Lemma-1
\begin{align*}
\frac{\sin(2nx)}{\sin^{2n}(x)}&=\...
0
votes
0
answers
132
views
Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
8
votes
2
answers
244
views
How to calculate $\int _0^1 \int _0^1\left(\frac{1}{1-xy} \ln (1-x)\ln (1-y)\right) \,dxdy$
Let us calculate the sum
$$
\displaystyle{\sum_{n=1}^{+\infty}\left(\frac{H_{n}}{n}\right)^2},
$$
where $\displaystyle{H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}}$ the $n$-th harmonic number.
My try
The ...
0
votes
1
answer
41
views
How to express $\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $ in terms of an integral?
I have this sum
$$\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $$
where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. I want to transform summation to an integral ...
- 51
0
votes
0
answers
33
views
Question on transforming a sum to an integral using the Euler–Maclaurin formula.
I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum
$$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+...
- 51
1
vote
0
answers
69
views
Leibniz integral rule for summation
Context
Fundamental points of Feymann trick:
You have an integral $I_0=\int_a^b f(t)\mathrm{d}t$
Now consider a general integral $I(\alpha)=\int_a^b g(\alpha,t)\mathrm{d}t$ so that $I'(\alpha)=I_0$ ...
1
vote
1
answer
95
views
Evaluation of $\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \log(\tan(\frac{\pi}{4} + x)))}{\tan(2x)} \,dx$ [closed]
$$\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \log(\tan(\frac{\pi}{4} + x)))}{\tan(2x)} \,dx$$
$$\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \...
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,
$$\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{2^{2n}}\binom{2n}{n}\sum_{k=1}^n (-1)^{k-1}\frac{H_k}{k}-\sum_{n=1}^{\infty}(-1)...
- 5,495
5
votes
2
answers
157
views
Show that $\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$
Show that $$\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$$
My try :
We know that
$$\sum_{n=1}^{\infty} \binom{2n}{n} (H_{2n} - H_{n}) ...
1
vote
2
answers
205
views
Which closed form expression for this series involving Catalan numbers : $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{4^nn^2}\binom{2n}{n}$
Obtain a closed-form for the series: $$\mathcal{S}=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{4^nn^2}\binom{2n}{n}$$
From here https://en.wikipedia.org/wiki/List_of_m ... cal_series we know that for $\...
6
votes
4
answers
241
views
Evaluate $\int_{0}^{1}\{1/x\}^2\,dx$
Evaluate
$$\displaystyle{\int_{0}^{1}\{1/x\}^2\,dx}$$
Where {•} is fractional part
My work
$$\displaystyle{\int\limits_0^1 {{{\left\{ {\frac{1}{x}} \right\}}^2}dx} = \sum\limits_{n = 1}^\infty {\...
user1235604
1
vote
3
answers
66
views
I want to use integration for performing summation in Algebra
I am a class 9th student. Sorry if my problem is silly.
I am trying to find the sum of squares from 1 to 10. For this I tried summation, and it was fine.
But now I came to know that Integration can be ...
0
votes
1
answer
81
views
Prove that $\int_0^1\lfloor nx\rfloor^2 dx = \frac{1}{n}\sum_{k=1}^{n-1} k^2$
First of all apologies for the typo I made in an earlier question, I decided to delete that post and reformulate it
I am asked to prove that
$$\int_{(0,1)} \lfloor nx\rfloor^2\,\mathrm{d}x =\frac{1}{n}...
- 131
3
votes
2
answers
206
views
why does $\pi$ always show up in $\int_0 ^1 \frac{x^c}{1+x^k} dx$ if $c\neq mk-1$ for all $m \in \mathbb{N}$
when I posted this question I was interested in the sum $ \sum_{n=0} ^{\infty} \frac{(-1)^n}{4n+3}$ but when I thought about the generalised sum $ \sum_{n=0} ^{\infty} \frac{(-1)^n}{kn+c +1}$ for all $...
- 6,565
5
votes
2
answers
185
views
Generating Function $\sum_{k=1}^{\infty}\binom{2k}{k}^{-2}x^{k}$
Closed Form For :
$$S=\sum_{k=1}^{\infty}\binom{2k}{k}^{-2}x^{k}$$
Using the Series Expansion for $\arcsin^2(x)$ one can arrive at :
$$\sum_{k=0}^{\infty}\binom{2k}{k}^{-1}x^{k}=\frac{4}{4-x}-4\arcsin\...
- 3,280
1
vote
0
answers
86
views
Prove that for any real numbers $x_1,\cdots, x_n, \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j\ge 0$
Let $D_1,\cdots, D_n$ be n disks in $\mathbb{R}^2$. For any $i,j,$ let $a_{ij}$ denote the area of $D_i\cap D_j.$ Prove that for any real numbers $x_1,\cdots, x_n, \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i ...
- 869
8
votes
1
answer
2k
views
Proving $\sum_{n=1}^{\infty}\binom{2n}{n}^2\frac{4H_{2n}-3H_n}{n2^{4n}}=\zeta(2)$
While trying to solve Prove $\int_{0}^{1}\frac1k K(k)\ln\left[\frac{\left(1+k \right)^3}{1-k} \right]\text{d}k=\frac{\pi^3}{4}$
I was able to reduce it to the following form,
$$\sum_{n=1}^{\infty}\...
- 3,280
1
vote
0
answers
60
views
Integration including the floor function
I'm aware of the way to quantify the following integral in the following way, however I'm trying to find another way to express the given integral. Especially when the function $f(x)$ is not ...
- 73
0
votes
1
answer
123
views
Is there an easy way to calculate this infinite summation
Is there an easy way to calculate this summation of integral:
$$\sum_{n=0}^\infty \int_{r=0}^1 \frac {(r-\frac{1}{2})\cos(c\cdot\ln(r+n))} {(r+n)^{1-b}} dr $$
The most obvious approach is to calculate ...
0
votes
0
answers
83
views
Re-writing a sum of binomial coefficients as an integral of shifted Legendre polynomials
This is a question regarding the answer presented here.
In order to make this post self-contained, I am wondering if someone can explain why the sum
$$
\sum_{k=0}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{...
user1211323
2
votes
3
answers
124
views
Evaluating $\int_{0}^{1}{\tan^{-1}(x)\, dx}$ via a series (not IBP)
I tried :
$$\begin{align}I&=\int_{0}^{1}{\tan^{-1}(x)\, dx}\\&=\int_{0}^{1}{\sum_{k\geq0}{\frac{(-1)^kx^{2k+1}}{2k+1}}\, dx}\\&=\sum_{k\geq0}{\frac{(-1)^k}{2k+1}\left[\frac{x^{2(k+1)}}{2(k+...
- 977
5
votes
2
answers
490
views
Evaluating $\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{ij(i+j)^2}$
I was playing around with double sums and encountered this problem: Evaluate
$$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{ij(i+j)^2}$$
It looks so simple I thought it must have been seen before, ...
- 840
3
votes
1
answer
129
views
Compact form of solution of $\displaystyle\int_0^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x$
I hope it won't be categorized as a trivial question, I solved this integral and arrived at the following form:
$${\int_{0}^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x=\frac{n\pi}{...
3
votes
0
answers
74
views
Function $\mathcal{Z}\left(x\right)=\sum_{n=1}^{\infty}n^{-x}\sin\left(n\right)$ [duplicate]
This is just a modification of the Zeta Function, if it is already present in literature, please link me to it.
$$\mathcal{Z}\left(x\right)=\sum_{n=1}^{\infty}n^{-x}\sin\left(n\right)$$
This is the ...
- 3,280
1
vote
0
answers
40
views
Is it possible to prove all main integral properties for the discreet sum integral formula?
There are a ton of different formulas for integrals, and my favourite is one I often hear called the discreet sum integral, defined such that
$$\int_a^bf(x)dx=\lim_{d\to0^+}\sum_{x=a/d}^{b/d}df(dx)$$
...
- 81
4
votes
2
answers
167
views
Summation $S=\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{k-1}}{k+\frac{1}{2}}\ln\left(1-\frac{1}{\left(k+\frac{1}{2}\right)^{2}}\right)\right)$
$$S=\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{k-1}}{k+\frac{1}{2}}\ln\left(1-\frac{1}{\left(k+\frac{1}{2}\right)^{2}}\right)\right)$$
Let,
$$f(a)=\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{...
- 3,280
1
vote
1
answer
153
views
Sum of values across a line?
I've been thinking recently and have confused myself a little bit as to the difference between sums and integrals.
I understand that an integral is a Riemann sum where you take the limit as the ...
- 13
7
votes
3
answers
292
views
Solve $\int_0^\infty\frac x{e^x-e^{\frac x2}}dx$
I was able to solve the integral $$\int_0^\infty\frac x{e^x-e^\frac x2}dx=4\left(\frac{\pi^2}6-1\right)$$ I want to see other approaches to solving it. Here is my solution: $$\int_0^\infty\frac x{e^x-...
- 6,549
1
vote
1
answer
47
views
Confused by a summation inside the integral: $\int_0^1 (v-u)^2 \sum_{i=0}^n \frac{u_i}{n}\delta_{i/n}(v) dv$
I came across the following integral:
$\int_0^1 (v-u)^2 \sum_{i=0}^n \frac{u_i}{n}\delta_{i/n}(v) dv$,
where the deltas represent the Dirac delta distribution. How would one evaluate this integral? I ...
- 585
2
votes
2
answers
172
views
how to turn the sum $\lim\limits_{n\to\infty}\left(\prod\limits_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$ into integral with Riemann sum
I want to solve
$\lim_{n\to\infty}\left(\prod_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$ with integration
I know that $\lim_{n \to \infty}\frac{b-a}{n} \sum_{k=0}^nf(a+\frac{b-a}{n})=\int_a^b f(x)dx$
so ...
- 6,565
7
votes
3
answers
277
views
Evaluating a Logarithmic Integral
For everything on this post $n$ and $m$ are positive integers.
The other day I found the following integral on the popular post "Integral Milking" and decided to give it a go.
$$\large\int_{...
2
votes
0
answers
84
views
Is it possible to compute this integral using this double summation?
I attempted to evaluate the following integral, $$\int_{0}^{1} \frac{\ln (1+x)}{1+x^2}\, dx$$
although I know it can be evaluated using Feynman's Technique of Integration, Trigonometric Substitution, ...
- 423
2
votes
0
answers
73
views
Sum with reciprocal of square central binomial $\sum\limits_{n = 1}^\infty {\frac{1}{{{n^3}{{\left( {C_{2n}^n} \right)}^2}}}}$
I am trying to evaluate this sum:
$$S=\sum\limits_{n = 1}^\infty {\frac{1}{{{n^3}{{\left( {C_{2n}^n} \right)}^2}}}}$$
My attempt:
$$\begin{array}{l}
S = \displaystyle\sum\limits_{n = 1}^\infty {\...
- 2,702
1
vote
0
answers
78
views
Generalization of $\displaystyle \int\frac{x^m}{p(x)}\mathrm{d}x$
How can this formula be generalized?
Let $p(x)\in\mathbb{R}[x]$ be a polynomial of degree $n$ with real or complex roots $z_1,...,z_n$ and $m\in\mathbb{N}$ such that $m=0,1,...,n-1$
$$\int\frac{x^m}{p(...
0
votes
1
answer
171
views
How to prove that $\sum_{k=1}^nf(k)>\int_1^nf(x)dx$
Context: This is a question I thought of when seeing a generalization of Euler's constant on its Wikipedia page.
We have a function $f:\mathbb{R}\mapsto\mathbb{R}$ which decreases as its input ...
- 6,549
0
votes
1
answer
246
views
Limits of Riemann sums left and mid endpoint rule.
In my calculus class we have begun talking about integrals. In particular we have begun talking about Reimann sums and how through the limit of a Reimann sum we can integral. But so far all our ...
- 11
0
votes
1
answer
98
views
Integral Identities for $S(x,y)=\sum_{k=1}^\infty\frac{y}{k^x(y+k)}$ and $\zeta(x)$ \ $\zeta(y)$
Define $$S(x,y)=\sum_{k=1}^\infty\frac{y}{k^x(y+k)}$$ This is a generalization of the harmonic function ($n=2$). There are many ways we could relate this to $\zeta(x)$. For example $$\lim_{y\...
- 6,549
2
votes
2
answers
68
views
How to show that a sum is inclosed by two values / bounds?
Show that this holds for all c>0
$\frac{\pi}{2\sqrt{c}} \le \sum_{n=0}^{\infty} \frac{1}{n^2 + c} \le \frac{\pi}{2\sqrt{c}} + \frac{1}{c}$
I'd really appreciate it if some can tell me if my proof ...
- 1,237
15
votes
2
answers
453
views
Closed form for $\int_0^1 \left\{ \frac{m}{x}\right\}^n\,dx$
I am trying to find the closed form for the following integral:
$$\int_0^1 \left\{ \frac{m}{x}\right\}^n\,dx, \quad \forall m,n \in \mathbb{N}$$
where $\{x\}$ denotes the fractional part of $x$. This ...
1
vote
1
answer
254
views
Prove that $\sum_{k=1}^n 1/k - \ln n \to \int_0^1 1/(1-x) + 1/(\ln x) dx$ as $n\to\infty$.
Prove that
$$
\sum_{k = 1}^{n}\frac{1}{k} - \ln\left(n\right) \to
\int_{0}^{1}\left[\frac{1}{1 - x} +
\frac{1}{\ln\left(x\right)}\right]{\rm d}x
\quad\mbox{as}\quad n\to\infty
$$
I don't think this ...
- 2,031
1
vote
0
answers
139
views
Solving a geometric-harmonic series
Find the value of $\displaystyle \frac21- \frac{2^3}{3^2}+ \frac{2^5}{5^2}- \frac{2^7}{7^2}+ \cdots$ till infinite terms.
found this problem while integrating $\arctan\left(x\right)/x$ from $0$ to $2$ ...
0
votes
0
answers
96
views
Finding out the value of $\sum_{n=1}^{\infty} \frac{1}{n} \sum_{k=n+1}^{\infty} \frac{(-1)^k}{(2k-1)^2} + 2\int_0^{\pi/4} \log^2(\cos x) dx$
We have to find the value of $$\sum_{n=1}^{\infty} \frac{1}{n} \sum_{k=n+1}^{\infty} \frac{(-1)^k}{(2k-1)^2} + 2\int_0^{\pi/4} \log^2(\cos x) dx$$
I have no idea where to even begin with, any help/...
- 121
0
votes
0
answers
46
views
Why can we approximate a sum by a definite integral?
From wikipedia https://en.wikipedia.org/wiki/Summation#Approximation_by_definite_integrals, I read that
$\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds$ for increasing ...
- 365
1
vote
1
answer
68
views
Prove that $-\int_0^1 (2(1-2x)^{2n} - 2)dx - \sum_{k=1}^n \int_0^1 {n\choose k} \dfrac{(-4x(1-x))^k}k dx = 2H_{2n} - H_n$
Prove that $-\int_0^1 (2(1-2x)^{2n} - 2)dx - \sum_{k=1}^n \int_0^1 {n\choose k} \dfrac{(-4x(1-x))^k}k dx = 2H_{2n} - H_n$ where $H_n$ is the nth Harmonic number.
For the first part, I think one can ...
- 2,031
1
vote
0
answers
62
views
Is it correct to evaluate these summations as integrals?
I have a question about this formula used to calculate the first critical speed of a drive shaft.
$$
n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}}→(1)
$$
It is the most commonly ...
- 55
0
votes
0
answers
56
views
prove that $\sum_{i,j=1}^n f(a_i - a_j) = \int_{-\infty}^\infty (\sum_{i=1}^n \frac{1}{1+(x-a_i)^2})^2 dx,$
Prove that if $a_1,a_2,\cdots, a_n$ are real numbers then $$\sum_{i,j=1}^n f(a_i - a_j) = \int_{-\infty}^\infty \left(\sum_{i=1}^n \frac{1}{1+(x-a_i)^2}\right)^2 dx,$$ where $f(y) = \int_{-\infty}^\...
- 1,137
3
votes
2
answers
154
views
Are there nice functions for which $\sum\limits_{n\geq 1} f(n) = \int\limits_{\mathbb{R}_+}f(t)dt$?
What can we say about the class of functions for which $$\sum\limits_{n\geq 1} f(n) = \int\limits_{\mathbb{R}_+}f(t)dt$$
Are there any good examples of such functions?
Edit: You may prefer different ...
- 95
1
vote
1
answer
147
views
Integral as weighted sum of derivatives. Is this a new result?
$$\int f(x) \, dx = \sum_{n=1}^\infty (-1)^{n+1}*\frac{x^n}{n!}\frac{d^nf(x)}{dx^n}$$
I derived this equation from the repeated application of the chain rule.
$$\int f(x) \, dx = \int 1*f(x) \, dx$$
$$...
- 11
0
votes
1
answer
65
views
Calculating the final sum of an investment with a specific daily growth of rate over a period of time.
Calculating the final sum of an investment with a specific daily growth of rate over a period of time.
I do apologize if this question is very basic for the vast majority of people in this forum but ...
- 3
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