All Questions
86
questions
4
votes
1
answer
91
views
Why $\infty=\sum_{i=1}^\infty \frac{1}{n+i}\neq\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$?
I was wondering why $\sum_{i=1}^\infty \frac{1}{n+i}$ diverges but $\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$. While assuming integral as limit of series, we find out that:
$$
\int_1^...
- 43
4
votes
1
answer
169
views
Prove that sum of integrals $= n$ for argument $n \in \mathbb{N}_{>1}$
ORIGINAL QUESTION (UPDATED):
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \...
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,
$$\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{2^{2n}}\binom{2n}{n}\sum_{k=1}^n (-1)^{k-1}\frac{H_k}{k}-\sum_{n=1}^{\infty}(-1)...
- 5,495
0
votes
0
answers
64
views
Converting complex-exponential summation to Fresnel integrals
I have a summation
$$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$$
when $N$ is sufficiently large. ...
- 71
0
votes
1
answer
81
views
Prove that $\int_0^1\lfloor nx\rfloor^2 dx = \frac{1}{n}\sum_{k=1}^{n-1} k^2$
First of all apologies for the typo I made in an earlier question, I decided to delete that post and reformulate it
I am asked to prove that
$$\int_{(0,1)} \lfloor nx\rfloor^2\,\mathrm{d}x =\frac{1}{n}...
- 131
3
votes
2
answers
181
views
Interchanging integration with summation on a specific example
Consider the infinite sum
$$S=\sum_{j=1}^\infty (-1)^{j +1}\frac{1}{2j+1} \int_0^1 \frac{1-x^{2j}}{1-x}dx$$
and the associated integral
$$L=\int_0^1 \sum_{j=1}^\infty\frac{ 1-x^{2j}}{1-x} (-1)^{j +1}\...
- 482
2
votes
1
answer
114
views
Complete the proof of $ \int_{-\infty}^{\infty}{e^{-x^2/2}x^{2n} dx} = \sqrt{2\pi} \frac{(2n)!}{2^nn!} $
What I already know is that if $L(s) := \int_{-\infty}^{\infty}{e^{sx}e^{\frac{-x^2}{2}} dx}=\sqrt{2\pi}e^{\frac{s^2}{2}}$, then $L^{(2n)}(0)=\int_{-\infty}^{\infty}{e^{\frac{-x^2}{2}}x^{2n} dx}$. So ...
- 65
1
vote
1
answer
74
views
Estimating a sum with an integral to nearest power of $10$
Suppose I want to estimate the sum $\sum_{x = -1000}^{1000} \sum_{y: x^2 + y^2 < 10^6 } x^2$, or identically, $\sum_{y = -1000}^{1000} \sum_{x: x^2 + y^2 < 10^6 } x^2$, to the nearest power of $...
- 731
1
vote
1
answer
52
views
A Riemann Sum clarification
I have to find the area under the curve $f(x) = x^2$ in the segment $[-2, 1]$ using Riemann Sum. So here is what I did, but it's wrong and I need some clarification about (see the questions in the end)...
- 3,482
2
votes
1
answer
59
views
Integration with Riemann Sum
I wanted to try to perform a Riemann Sum for the following integral, but I got stuck in the middle.
$$\int_{-1}^0 e^{-x^2}\ \text{d}x$$
So the interval is $[-1, 0]$, and I chose $\Delta x = \dfrac{1}{...
- 3,482
0
votes
0
answers
46
views
Why can we approximate a sum by a definite integral?
From wikipedia https://en.wikipedia.org/wiki/Summation#Approximation_by_definite_integrals, I read that
$\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds$ for increasing ...
- 365
1
vote
1
answer
97
views
How to evaluate $\sum_{k=0}^{\infty} \frac{(-1)^{k}(2k+1)!!}{(k+1)(2k+2)!!}\frac{\pi}{2}\alpha^{k+1}$
I was trying to solve the integral $\int_0^{\frac{\pi}{2}} \ln(1+\alpha\sin^2 x)\, dx$, where $\alpha>-1$
And my approach is using infinite sum by expanding $\ln(1+\alpha\sin^2 x)$
So let $I=\int_0^...
- 858
4
votes
2
answers
202
views
Evaluate the following humongous expression
PROBLEM:
Evaluate $$\left(\frac{\displaystyle\sum_{n=-\infty}^{\infty}\frac{1}{1+n^2}}{\operatorname{coth}(\pi)}\right)^2$$
CONTEXT:
I saw a very interesting and yet intimidating question on the ...
- 1,253
0
votes
0
answers
56
views
prove that $\sum_{i,j=1}^n f(a_i - a_j) = \int_{-\infty}^\infty (\sum_{i=1}^n \frac{1}{1+(x-a_i)^2})^2 dx,$
Prove that if $a_1,a_2,\cdots, a_n$ are real numbers then $$\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,$$ where $f(y) = \int_{-\infty}^\...
- 1,137
1
vote
0
answers
71
views
Evaluate $\int_1^e\frac{1}{x}dx$ using upper and lower sums
Using the upper and lower sums, I've tried to solve $\int_1^e\frac{1}{x}dx$ in the following way.
Let $$\Delta x = \frac{\mathit e -1}{n}$$ and let a partition $P$ be given by $$ P = \{1,1+\Delta x, 1+...
- 11
6
votes
4
answers
500
views
Finding $\sum_{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$
I want to find the closed form of:
$\displaystyle \tag*{} \sum \limits _{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$
Where $H_{k}$ is $k^{\text{th}}$ harmonic number
I tried to ...
- 921
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:
$$\displaystyle \tag*{} \sum \limits_{n=1}^{\infty} \dfrac{2(2n+1)}{\exp\left( \dfrac{\pi(2n+1)}{2}\right)+\exp \left( \dfrac{3\pi(2n+1)}{2}\right)}...
- 921
1
vote
1
answer
115
views
Why is this sum equal to an integral?
I participate the stochastic course and we now speak about summable families. There we have the following definition:
Let $\Omega$ be countable and $a:\Omega\rightarrow \Bbb{R}_+\cup \{\infty\}$ be a ...
- 2,935
0
votes
1
answer
84
views
Inequality involving finite sum and integral
I'm reading a proof where they use the following inequality:
$$\sum_{k=4}^n \frac{k^2}{n}(1-a)^{k+1}\le\int_0^\infty \frac{x^2}{n}\exp{(-ax)}$$
For $a>0$. I'm trying to show it.
So far I got
$$\...
- 376
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
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$$S=\sum_{i=0}^\infty \frac{1}{(4i+1)(4i+2)(4i+3)} =\sum_{...
- 9,599
6
votes
1
answer
260
views
Formula for $f(1) + f(2) + \cdots + f(n)$: Euler-Maclaurin summation formula
Let $f\colon \mathbb{R}\to \mathbb{R}$ be a function with $k$ continuous derivatives. We want to find an expression for
$$
S=f(1)+f(2)+f(3)+\ldots+f(n).
$$
I'm currently reading Analysis by Its ...
- 435
2
votes
1
answer
230
views
Unable to solve integral of a summation
I want to integrate each datapoint in a 2D image $X$ over a disc, but i'm really unsure what to do with the summation term. Essentially, at each point $x_j$, we integrate a disc of radius $r$, ...
- 73
1
vote
2
answers
201
views
Determine if $\int_1^\infty \arctan(e^{-x})dx$ converges or diverges
I want to check if the sequence $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})]$ converges absolutely, by condition, or not.
We know that $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})] \le \...
- 627
2
votes
1
answer
45
views
Replacement of the tagged partition displacement of the Riemann Integral
In specific, my question is if I can make the following by the definition on Riemann integral...
$$\int_a^bf(x)dx = \lim_{\lambda \rightarrow 0}\sum_{i=1}^n f(c_i)\Delta x_i = \lim_{\lambda \...
- 59
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 $$y[m]=x[n]*h[n] = \sum_{l=-\infty}^{+\infty}x[l]h[m-l] = \cases{\frac{27\times2^{3 - m}}{5} \ m \ge 2 \\ \frac{...
- 4,359
0
votes
1
answer
73
views
Integration and summation inequality
This step arise while proving infinite Hilbert matrix is bounded linear operator using Schurz test.
In my book (Functional Analysis by S Kesavan) it's given as follows:
$$ \sum_{i=0}^\infty \frac{1}{(...
- 2,506
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) = ...
0
votes
1
answer
216
views
Is it possible to interchange sum and integral when the series doesn't converge uniformly?
Suppose a series $\sum f_n$ of integrable functions on $[0,1]$ does not converge uniformly, is it possible that $$\sum_{n=1}^\infty \int_0^1 f_n(x)dx = \int_0^1\sum_{n=1}^\infty f_n(x)dx$$ still holds?...
- 225
1
vote
3
answers
71
views
Bounding sum by (improper) integral
I am trying to verify the following inequality that I came across while reviewing some analysis exercises online:
$$
\sum_{n=1}^{k} \left(1-\frac{n}{k}\right)n^{-1/7}\leq \int_{0}^{k}\left(1-\frac{x}{...
- 35
5
votes
1
answer
136
views
If $|f'(c)|<M$, prove $|\int_{0}^{1}f(x)dx-1/n \sum_{k=0}^{n-1}f(x/n)|<M/n$ [duplicate]
We have a derivative function $f$ with for every $c$ element of $\mathbb{R}: |f'(c)|<M$. I tried to prove that
prove that $\displaystyle \left|\int_{0}^{1}f(x)\mathrm{d}x-\frac{1}n \sum_{k=0}^{n-1}...
- 875
11
votes
3
answers
449
views
Does parity matter for $\lim_{n\to \infty}\left(\ln 2 -\left(-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots -\frac{(-1)^n}{n}\right)\right)^n =\sqrt{e}$?
Prove that $$\lim_{n\to \infty}\left(\ln 2 -\left(-\frac12+\frac13-\frac14+\cdots -\frac{(-1)^n}n\right)\right)^n =\sqrt{e}$$
I happened to encounter this problem proposed by Mohammed Bouras,...
- 3,442
1
vote
3
answers
112
views
Summation coincides with integration?
Let $a,b \in \mathbb{N}$ such that $a<b$.
Then let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function on the interval $[a,b]$.
Under what conditions the following property is satisfied :
$$\...
- 23
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
8
votes
4
answers
352
views
A double sum for the square of the natural logarithm of $2$.
I am trying to show
\begin{eqnarray*}
\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{(n+m)^2 2^{n}} =(\ln(2))^2.
\end{eqnarray*}
Motivation : I want to use this to calculate $ \operatorname{Li}_2(...
- 36.8k
1
vote
3
answers
328
views
Prove that $\sum_{k=1}^{2n} \frac{(-1)^{k-1}}{k} = \sum_{k=1}^n \frac{1}{n+k}$ [duplicate]
I am trying to prove that $$\sum_{k=1}^{2n} \frac{(-1)^{k-1}}{k} = \sum_{k=1}^n \frac{1}{n+k}$$ for $n \in \mathbb{N}$.
My approach is to prove this by induction and this is what I got so far:
$$\...
- 135
5
votes
2
answers
878
views
Swapping integral and sum using dominated convergence theorem
Show that
$$
\sum_{n=1}^\infty \int_0^\infty t^{s/2-1}e^{-\pi n^2t}dt
= \int_0^\infty t^{s/2-1} \sum_{n=1}^\infty e^{-\pi n^2t} dt
$$
with $s > 1$ using the dominated convergence theorem.
If ...
- 450
1
vote
2
answers
115
views
Switching the Order of Summation and Integration: Incomplete Gamma Function
I want to solve the following integral:
$$ I(x) = \int_0^\infty \gamma(N, x) f(x) \, dx $$
where $|f(x)| \leq 1$ and $\gamma(N, x)$ is the incomplete Gamma function with $N \in \mathbb{Z}_{++}$. To ...
- 1,079
0
votes
1
answer
69
views
Explanation of Replacing Summation and Integration
Please explain the below line in detail -
$$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}- \int_{1}^{\infty}\frac{1}{x^s}dx= \sum_{n=1}^{\infty} \int_{n}^{n+1} \left(\frac{1}{n^s}-\frac{1}{x^s}\...
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:
$$ \displaystyle \mathcal{I}=\int_{0}^{\infty}\frac{\sin(x)}{x}\DeclareMathOperator{\dilog}{Li_{2}}\dilog\left(-e^{-x}\right)\...
0
votes
2
answers
130
views
Proving integral $\int_0^1\frac{e^x-1}{x}$ is equal to $\sum_{n=1}^{\infty}\frac{1}{n \cdot n!}$
Show that the following equality is true.
$$
\int_0^1\frac{e^x-1}{x}\, \mathrm dx = \sum_{n=1}^{\infty}\frac{1}{n \cdot n!}
$$
How can I tackle this problem?
- 2,190
0
votes
1
answer
579
views
Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties
For function defined as
$$
f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}
$$
check if $f$ is continuous and differentiable function.
My approach:
I would like to use the connection between this sum and ...
- 11
9
votes
1
answer
807
views
Evaluating the limit of a sum using integration
One of the first results we learn in definite integral is that if $f(x)$ is Riemann integrable in $(0,1)$ then we have $\lim_{n \to \infty}\dfrac{1}{n}\sum_{i=1}^{n}f\Big(\dfrac{i}{n}\Big) = \int_{0}^{...
- 20.8k
1
vote
1
answer
45
views
Pattern matching in an indefinite integral over a sum involving Gamma functions and exponentials.
Suppose that we have the following fact: For $h > 0$,
$$\int_0^{\infty} z^{-\frac{3}{2}} \sum_{n=0}^{\infty} (-1)^n \frac{2^h}{\Gamma(h)} \frac{\Gamma(n+h)}{\Gamma(n+1)} \frac{(2n+h)}{\sqrt{2\pi}} \...
- 189
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
2
votes
2
answers
319
views
Changing order of partial sum and integral all under limit to infinity
$$ \lim_{n \rightarrow \infty} \int_{a}^b\ \sum_{k=1}^{n}f_k(x) \mathrm dx= \sum_{k=1}^{\infty}\int_{a}^b f_k(x) \mathrm dx $$
Is this generaly true ? Integral is a sum , two sums can interchange, ...
- 1,641
0
votes
0
answers
52
views
How to make a multiple integration by parts rigourous
Sometimes, in order to prove an identity about the integral of a function we can apply multiple time integration by parts until we integrate an easy function.
For example it's a nice technique with ...
- 666
0
votes
1
answer
80
views
Riemann sums over dense countable sets
Let $f$ and $g$ be positive, smooth and integrable functions in $\mathbb{R}$, whose derivatives are also integrable.
Assume as well that the expression
$$
\frac{\sum_{q\in \mathbb{Q}} f(q)}{\sum_{q\in ...
- 1,859
3
votes
1
answer
248
views
Examples where we cannot interchange summation and integration
We know of conditions under which we can interchange summation and integration (1, 2).
What are some simple examples where we cannot do so and which we could present to high-school/introductory ...
user547493
1
vote
2
answers
93
views
A relation involving summation and integration
Let, $\displaystyle C(x)=\sum_{n\le x}c_n$, where $\{c_n\}$ is a sequence of complex numbers ; and let $f(t)$ be a continuously differentiable function such that $\displaystyle \lim_{Y\to \infty}C(Y)f(...
- 772