Tagged Questions
2
votes
2answers
45 views
Calculate the length of curve $f(x)=\arcsin(e^x)$, check solution, please.
As in the topic, my task is to calculate the length of $f(x)=\arcsin(e^x)$ between $-1, 0$. My solution: I use the the fact, that the length of $f(x)$ is equal to $\int_{a}^b\sqrt{1+(f'(x))^2}dx$ ...
2
votes
1answer
42 views
Closed Form of a Particular Sum
Does anyone have any ideas on how to find a closed form for the following expression? It comes up when trying to bound a particular integral. The sum is:
$$\sum_{n=0}^{\infty} ...
4
votes
1answer
113 views
Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
By integral test, it is easy to see that
$$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$
converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$]
I am ...
0
votes
0answers
72 views
Simplifying $\frac{1}{n}\sum_{k=1}^n f(\frac{1}{k})$
Suppose that $$\displaystyle \forall x\in \mathbb{R}_+^* \quad f(x)=\frac{x^2-1}{4}-\frac{\ln(x)}{2}.$$
How can I simplify this: $$I(n)=\frac{1}{n}\sum_{k=1}^n f\left(\frac{1}{k}\right)$$
and prove ...
1
vote
1answer
31 views
Show the infinite sum is an integral
Here is a prelim question that I have not been able to solve:
Show that if $a>1$, then
$$
\sum_{n=1}^\infty n^{-1/2}a^{-n} = 1/\sqrt{\pi}\int_0^\infty \frac{y^{-1/2}}{ae^y-1}dy
$$
Thanks so ...
2
votes
2answers
48 views
Show convergence and calculate the limit
I guess it has something to do with riemann sums but this is new for me.
$\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$
How do i start?
0
votes
0answers
18 views
Lower Bound for the Thomson Problem
In a paper I am reading it gives the following:
$$E_{-1}(\omega_N) \geq -c(-1,3) \cdot N^{3/2}$$
$$E_{-1}(\omega_N) = \sum_{j\neq k}^N \left( \frac{1}{|x_j-x_k|} -m(-1,3) \right)$$
$$ \sum_{j\neq ...
2
votes
1answer
61 views
How to convert from Riemann sum to integral?
Im converting this to integral: But I need help!
$$\sum_{i=1}^4{\left(-2+i\frac12\right)^3*\frac12}$$
$$\sum_{i=1}^4{\left(\frac{-13}2+\frac{47i}8\right)\frac12}$$
$$\Delta ...
4
votes
1answer
51 views
Limit of difference of integral and sum
$f:[0,1]\rightarrow\mathbb R$ and $f\in C^1$, then the limit $\lim_{n\rightarrow\infty} n(\int_{0}^{1}f(x)dx-\frac{1}{n}\sum_{k=1}^{n}f(\frac{k-1}{n}))$ exists.
I guess the kernel lies in the sum ...
4
votes
2answers
54 views
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, $n(r) = \#\{\alpha_j \le r\}$. Prove $\int_0^1n(r)dr = \sum_{j=1}^\infty(1-|\alpha_j|)$.
I'm trying to solve the following exercise from chapter 15 of Rudin's Real and Complex Analysis:
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, and let $n(r)$ be the number of terms in the ...
3
votes
2answers
85 views
Convergence of integral of log and sum the mean of the logs
How can I show that the following limit converges and $L \in (0, +\infty)$?
$\lim\limits_{n \to +\infty}\left( S_n - T_n\right)$, where $S_n = \int\limits_1^n \log x\, dx$, and $T_n = \sum_{k = ...
1
vote
1answer
69 views
How to find discrete integral given different time intervals.
I want to implement a PID controller and I'm unsure of how to find the integration part. Normally the integral is calculated as $\sum_{n=1}^{t} e_{n}$, where $e_n$ is the sample error at time n. ...
5
votes
1answer
65 views
A sum refers to euler's constant
Show that :
$$\sum\limits_{m=1}^{\infty }{\sum\limits_{n={{2}^{m-1}}}^{{{2}^{m}}-1}{\frac{m}{\left( 2n+1 \right)\left( 2n+2 \right)}}}=1-\gamma $$
14
votes
1answer
437 views
A nice log trig integral
Show that :
$$\int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\cos x{{\ln }^{2}}\sin x}{\cos x\sin x}}\text{d}x=\frac{1}{4}\left( 2\zeta \left( 5 \right)-\zeta \left( 2 \right)\zeta \left( 3 \right) ...
1
vote
0answers
23 views
Quick question on the simplification of digamma series
How to simplify :
$$\sum\limits_{k=1}^{\infty }{\frac{{{\left( -1 \right)}^{k-1}}}{k}\left( \psi \left( \frac{k}{2\left( 2+\sqrt{3} \right)}+1 \right)-\psi \left( \frac{k}{2\left( 2+\sqrt{3} ...
2
votes
1answer
71 views
Integral inequality with sin exp
For $\displaystyle f(x)=\int_x^{x+1}\sin (\text{e}^t)\text{d}t$
Prove that :
$\displaystyle \text{e}^x\left|f(x)\right|\le 2$
2
votes
1answer
82 views
A integral equation
Prove that :
$$\displaystyle \int_0^{\frac{\pi}{2}}p(x)\cot x\text{d}x=2\sum_{k=1}^{\infty}\int_0^{\frac{\pi}{2}}p(x)\sin (2kx)\text{d}x$$
where $\displaystyle p(x)=x^n$
4
votes
3answers
115 views
Help in manipulating Integrals
I try to express : $\displaystyle 1+2\sum _{ k=1 }^n \cos(2k\theta ) $
as : $\dfrac { \sin\left( \theta +2\theta n \right) }{ \sin\left( \theta \right) } $
I tried to use the exponential function ...
13
votes
2answers
142 views
Evaluate $\sum_{n=1}^\infty 1/n^2$ using $\int_0^1 \int_0^1 \frac{\mathrm{d}x \, \mathrm{d}y}{1-xy}$
This paper http://math.ucsb.edu/~cmart07/Evaluating%20Integrals.pdf hints at a way to compute the sum $$ \sum_{n=1}^\infty \frac{1}{n^2} $$ by expanding it into the double integral $$\int_0^1 \int_0^1 ...
2
votes
2answers
209 views
How to prove this sum of integrals
How to prove this ?
$$\displaystyle \sum\limits_{k=0}^{\infty }{\int_{2k\pi }^{\left( 2k+1 \right)\pi }{{{\text{e}}^{-\frac{x}{2}}}\frac{\left| \sin x-\cos x \right|}{\sqrt{\sin ...
10
votes
1answer
359 views
How to evaluate $\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$
I want to evaluate $$\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$$
I run this integral on Maple, It does converge. How we get a closed form?
Is that ...
1
vote
2answers
214 views
how to determine variable n when not given in Simpson's Rule problem
I have $$\int_0^2 \frac{1}{1+x^6} dx$$
We vaguely went over this in class today and I understand how to work Simpson's rule but only when I have $n$ already given to me. How can I determine $n$ when ...
6
votes
2answers
141 views
Evaluate $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2}$
Considering the sum as a Riemann sum, evaluate $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2} .$$
5
votes
2answers
115 views
Simplifiying sum through integral?
I wanted to compute the sum $$\sum_{k=1}^{n}\frac{1}{k}\binom{n}{k}.$$
And I thought it would be easiest to do this by making it a function, differentiating it and integrating it then.
So I did: ...
5
votes
2answers
173 views
Math Courses involving clever integration techniques
I am a third year undergraduate mathematics student.
I learned some basic techniques for simplifying sums in high school algebra, but I have encountered some of the more interesting techniques in my ...
8
votes
3answers
226 views
sum of a series
Can
\begin{equation}
\sum_{k\geq 0}\frac{\left( -1\right) ^{k}\left( 2k+1\right) }{\left(
2k+1\right) ^{2}+a^{2}},
\end{equation}
be summed explicitly, where $a$ is a constant real number? If $a=0,$ ...
4
votes
1answer
163 views
Computing a finite binomial sum
I want to compute $$S(n,m,a)=\sum_{k=0}^{n}k^{m}\cdot\binom{n}{k}\cdot a^k.$$
With $n,m\in\mathbb N$, $a\neq0$ and $S(n,0,a)=(a+1)^n$.
What I have found already:
I don't see any other options then ...
1
vote
2answers
348 views
Find the limit by expressing it as a definite integral of an appropiate function via Riemann Sums.
Find the limit $$\lim_{n\to \infty}\sum_{i=1}^{n}\frac{i}{n^2+i^2}$$ by expressing it as a definite integral of an appropiate function via Riemann Sums.
Observation: $n$ must refer to the number ...
