All Questions
86
questions
58
votes
7
answers
25k
views
Is it possible to write a sum as an integral to solve it?
I was wondering, for example,
Can:
$$ \sum_{n=1}^{\infty} \frac{1}{(3n-1)(3n+2)}$$
Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals.
But ...
- 11.2k
32
votes
1
answer
1k
views
Geometric representation of Euler-Maclaurin Summation Formula
In Tom Apostol's expository article (here's a free link), upon seeing the figure below (or this from the Wolfram project) I was expecting more diagrams to come to continue the error decomposition of ...
- 7,093
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
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$$S=\sum_{i=0}^\infty \frac{1}{(4i+1)(4i+2)(4i+3)} =\sum_{...
- 9,599
9
votes
1
answer
807
views
Evaluating the limit of a sum using integration
One of the first results we learn in definite integral is that if $f(x)$ is Riemann integrable in $(0,1)$ then we have $\lim_{n \to \infty}\dfrac{1}{n}\sum_{i=1}^{n}f\Big(\dfrac{i}{n}\Big) = \int_{0}^{...
- 20.8k
8
votes
3
answers
398
views
Evaluate $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2}$ [duplicate]
Considering the sum as a Riemann sum, evaluate $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2} .$$
- 6,510
8
votes
4
answers
352
views
A double sum for the square of the natural logarithm of $2$.
I am trying to show
\begin{eqnarray*}
\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{(n+m)^2 2^{n}} =(\ln(2))^2.
\end{eqnarray*}
Motivation : I want to use this to calculate $ \operatorname{Li}_2(...
- 36.8k
6
votes
4
answers
500
views
Finding $\sum_{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$
I want to find the closed form of:
$\displaystyle \tag*{} \sum \limits _{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$
Where $H_{k}$ is $k^{\text{th}}$ harmonic number
I tried to ...
- 921
6
votes
1
answer
260
views
Formula for $f(1) + f(2) + \cdots + f(n)$: Euler-Maclaurin summation formula
Let $f\colon \mathbb{R}\to \mathbb{R}$ be a function with $k$ continuous derivatives. We want to find an expression for
$$
S=f(1)+f(2)+f(3)+\ldots+f(n).
$$
I'm currently reading Analysis by Its ...
- 435
6
votes
0
answers
535
views
Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$
Hi I am trying to integrate and obtain a closed form result for
$$
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.
$$
Here is what I tried (but I do not think this is ...
- 9,966
5
votes
1
answer
136
views
If $|f'(c)|<M$, prove $|\int_{0}^{1}f(x)dx-1/n \sum_{k=0}^{n-1}f(x/n)|<M/n$ [duplicate]
We have a derivative function $f$ with for every $c$ element of $\mathbb{R}: |f'(c)|<M$. I tried to prove that
prove that $\displaystyle \left|\int_{0}^{1}f(x)\mathrm{d}x-\frac{1}n \sum_{k=0}^{n-1}...
- 875
5
votes
2
answers
878
views
Swapping integral and sum using dominated convergence theorem
Show that
$$
\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
$$
with $s > 1$ using the dominated convergence theorem.
If ...
- 450
5
votes
2
answers
142
views
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, $n(r) = \#\{\alpha_j \le r\}$. Prove $\int_0^1n(r)dr = \sum_{j=1}^\infty(1-|\alpha_j|)$.
I'm trying to solve the following exercise from chapter 15 of Rudin's Real and Complex Analysis:
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, and let $n(r)$ be the number of terms in the ...
- 5,487
5
votes
0
answers
293
views
If $f$ is integrable, then $\sum\limits_{n\ge 1}\frac{1}{\sqrt n}\vert f(x-\sqrt n)\vert$ is almost everywhere finite
I would like to show that $$\sum_{n\ge0}\left\vert \frac{1}{\sqrt n} f \left(x-\sqrt n \right)\right\vert \tag{$*$}$$ converges for almost every (a.e.) $x$.
The only technique I have is based on the ...
- 3,918
4
votes
1
answer
91
views
Why $\infty=\sum_{i=1}^\infty \frac{1}{n+i}\neq\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$?
I was wondering why $\sum_{i=1}^\infty \frac{1}{n+i}$ diverges but $\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$. While assuming integral as limit of series, we find out that:
$$
\int_1^...
- 43