6
votes
4answers
763 views

What's the purpose of this formula?

Just found this image on the web: Can anyone explain what's the meaning (if any) of this formula? (I did a Google image search but found no answer)
4
votes
6answers
104 views

If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous
0
votes
4answers
135 views
0
votes
1answer
27 views

Double integral via Riemann sum

How do I integrate the function $f(x,y)=15(x^{2}+y^{2})$, in $Q=[0,1]\times[0,1]$ via Riemann sum? I tried to get the partition $$0=x_{0}<x_{1}<\ldots<x_{n}=1\quad\text{and}\quad ...
0
votes
1answer
99 views

How does one graph $\sum_{x=0}^{n}$ [closed]

How does one graph a summation, like $$\sum_{x=0}^{n} n$$ Can it be like this Because if you take the points from the summation (0,0), (1,1), (2,3), (3,6) you can tell by summations it only works ...
1
vote
1answer
37 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
2
votes
4answers
115 views

Fastest way to integrate $\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $

This integral looks simple, but it appears that its not so. All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :)
1
vote
1answer
45 views

Is an integral without a differential component on a finite number of points just a sum?

Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ...
-1
votes
1answer
66 views

Rewrite and approximate the sum as an Integral $\sum_{i=1}^{1000} \sqrt{i}$ [closed]

This is not an Infinite sum !, how do we change this to an Integral. $ $ We normally write an integral as an infinite sum.
0
votes
2answers
31 views

If a sequence $\{a_n\}$ satisfies the Inequality $a_{n+1} < ka_{n}$, then show that $ \lim\limits_{n \to \infty} a_n =0$ where $0< k , a_n< 1$

I know one solution. Consider $\sum a_n$ Then use ratio test to show that the series converges, hence the sequence. Any other Ideass !
1
vote
1answer
47 views

Finding the limit of an integral

Evaluate $$\displaystyle\lim_{j\rightarrow \infty} \displaystyle\int_{0}^{a} \frac{1}{j!} \left(\ln \left(\frac{A}{x}\right)\right)^{j}dx$$
1
vote
1answer
45 views

$\sum_{x=a}^{b-1}\frac{1}{x}$ and $\sum_{x=a+1}^b\frac{1}{x}$

I have to prove the following relations: $\sum_{x=a}^{b-1}\frac{1}{x}\geq\log b - \log a $ $\sum_{x=a+1}^{b}\frac{1}{x}\leq\log b - \log a $ I tried to use the relation that $\int_a^b \frac{1}{x} ...
0
votes
0answers
34 views

What are the advantages/disadvantages of integration vs. summation?

If we are given a function, $f(x)$, we can either integrate it or sum it. I'm wondering what integration can do with $f(x)$ that summation can't, and what summation can do that integration can't. ...
1
vote
1answer
38 views

Can we possibly exchange summation and integration with negative values?

This is an attempt to go further than this answer. Essentially, we have either a summation of an integral: $$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$ ...or an integral of a summation: ...
2
votes
1answer
63 views

When can we use substitution for both integrals and summations?

This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?". Essentially, I would like to know, if we have: ...
0
votes
1answer
31 views

Formula for area under the curve

I don't know that the equation that I am going to explain below is correct or not, and this is why I am asking this question. So, I have found out that area under the curve could be found out by ...
0
votes
1answer
68 views

Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
4
votes
0answers
137 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 ...
3
votes
1answer
50 views

Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
1
vote
0answers
41 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
2
votes
1answer
80 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
2
votes
1answer
51 views

What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
2
votes
1answer
80 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
0
votes
0answers
74 views

Is this proof correct?

I was working on a summation problem, and I thought of a way to solve it. This is a proof of a generalisation of that method, is it correct, can it be improved? We will describe the conditions ...
0
votes
1answer
45 views

compare summation and integral

If we know $f(x)$ is monotonic decreasing on the interval $a \leq x < \infty$, could we obtain following relation formally? \begin{equation} \int_{m}^{n+1} f(x) dx \leq \sum_{j=m}^n f(j) \leq ...
0
votes
2answers
42 views

Prove a inequality about integral and summation

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally? \begin{equation} f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) ...
2
votes
3answers
76 views

Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques

I can't seem to use certain methods such as $\varepsilon$-N, L'Hôspital's Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler's Integral Representation to prove that ...
0
votes
1answer
37 views

Evaluating the following sum

I have no idea how to solve evaluate this integral: $$\lim_{n\to\infty} \frac{1^a + 2^a + \cdots + n^a}{n^{1+a}}, a > -1$$ I want to set this up as some sort of integration since it is a ...
1
vote
1answer
50 views

Proofs of $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$ [duplicate]

Prove that $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$. This comes from this question Evaluate $\sum_0^\infty \frac{1}{n^n}$.
0
votes
0answers
43 views

How to solve summation problem?

I want to evaluate " sum f(x) g'(x)" the limits of sum goes from 0---> infinity. I know how to solve this problem using integration by substitution method. I also know that using Riemann sum we can ...
1
vote
1answer
80 views

How do I express the integral of $\tan^{2n}(x)$ using sigma notation?

How do I express the integral of $\tan^{2n}(x)$ with respect to $x$ using sigma notation? $$\int \tan^{2n}(x)\, dx = \text{???}$$
0
votes
2answers
81 views

$\sum_{i=1}^n 1/i \leq c\log n$

This is what I want to show: $\sum_{i=0}^n 1/i \leq c \log n$ for all $n>N$ My current approach was this: $\sum_{i=1}^n 1/i = ( \int \sum_{i=1}^n 1/i )' = ( \sum_{i=1}^n \int 1/i )' = ( ...
0
votes
1answer
138 views

Find an expression for the area under the graph of f(x) as a limit?

f(X) = 2x/(x^2 +1), 1 <= x <= 3 Basically, I need to find an expression for the area under the graph within these intervals for the function as a limit. I understand the concept of the area ...
2
votes
0answers
42 views

calculating sum of a limit of integral

I am trying to calculate the following expression $$ \sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy ...
4
votes
2answers
269 views

maybe this sum have approximation $\sum_{k=0}^{n}\binom{n}{k}^3\approx\frac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty$

prove or disprove this $$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$ this problem is from when Find this limit ...
0
votes
2answers
170 views

Difference between summation and integration

It is well known that if a series $\sum\limits_{k= 0}^\infty a_k$ converges, then $a_k \to 0$. However, this is not true for integrals. What makes them different? Is it simply that they are ...
0
votes
1answer
44 views

Integrating over sums.

I want to make an integral. I know that Integral and Sum can be exchanged. But if I have the following case? $$ \int\left(\sum_{i_1}\sum_{i_2}e^{ix}\right)\sum_{j_1}\sum_{j_2}e^{jx}\,\text dx $$ ...
2
votes
2answers
101 views

Finding the complex fourier series of the function $x^2sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
0
votes
1answer
29 views

integral alternative of $\sum f(p)$ from $(1<\text{all primes}\leq n)$ to $(\text{maximum prime}<n)$

I have a naive question that if someone could find the integral alternative of $$\sum_{\substack{2\le p\le n\\p\text{ is prime}}} f(p)$$ where f(p) is a non-decreasing monotonic function.
0
votes
2answers
107 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
2
votes
1answer
43 views

When can we interchange Fourier transform and countable sum?

When does $\mathcal{F}\left ( \sum_{n=1}^{\infty} f_n (x)\right ) = \sum_{n=1}^{\infty} \mathcal{F}(f_n(x))$ where $\mathcal{F}$ the Fourier transform operator.
8
votes
0answers
134 views

Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
2
votes
2answers
77 views

How to interpret this formula from dsp? [closed]

Can you describe me what this formula does and what result in the end? $$\delta(x)=\begin{cases} 0, & \text{if $x \neq 0$} \\ 1, & \text{if $x = 0$} \\ \end{cases}$$ ...
2
votes
0answers
35 views

Show that $ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4}$ [duplicate]

Show that: $$ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4}$$ My idea: I thought that this could be rewritten as an integral, then use trig substitution (perhaps tangent) and then ...
3
votes
2answers
173 views

$\int_0^{\frac{\pi}{2}}x\cot(x)dx$ and $ \lim_{m \rightarrow \infty}\log\left( e^{2m}\left(\frac{(2m-1)!!}{(2m+1)^m}\right)^2\right)$.

I'm trying to evaluate the integral, but in doing so have stumbled upon the limit, which I don't know whether it exists, and if so how to resolve it (and whether I've derived the relationship between ...
0
votes
2answers
94 views

Prove that $\lim\limits_{n\rightarrow\infty} \displaystyle\sum\nolimits_{k=1}^n\frac{n}{n^2+k^2x^2} =\frac{\tan^{-1}(x)}{x}$ [closed]

Prove that $$\lim\limits_{n \to + \infty} \left( \sum\limits_{k=1}^n \frac{n}{n^2+k^2x^2} \right) =\frac{\tan^{-1}(x)}{x}.$$
3
votes
2answers
43 views

Find the antiderivative of a function with a finite series and factorials

If $n\in\mathbb{N},s\leq n$, I know that $$ \int_0^1 t^s(1-t)^{n-s-1}dt=\frac{s!(n-s-1)!}{n!}. $$ I would like to find a similar formula: is there a function $f(t)$ such that $$ \int_0^1 f(t) ...
2
votes
1answer
50 views

How $\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!}=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$

$$P(X\geq m)=\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!};m=0,1,...$$ Show that for any $m=1,2,...$ $$P(X\geq m)=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$$ I couldn't derive it also ...
4
votes
2answers
109 views

Finding non-trivial functions $f(x)$ such that $\sum_{n=1}^{\infty}f(n)=\int_{0}^1f(x)dx$

It is known that Atle Selberg found the following expression when he was 14 years old : $$\sum_{n=1}^{\infty}n^{-n}=\int_{0}^1x^{-x}dx.$$ Then, here is my question. Question : Find the other ...
1
vote
2answers
148 views

Can I get a little help proving equality between a summation and integral?

Prove$$\sum_{k=0}^x \binom{n}{k}p^{k}(1-p)^{n-k} =(n-x)\binom{n}{x}\int_{0}^{1-p}t^{n-x-1}(1-t)^{x}dt.$$ Can someone show me the steps please? Here is the hint my book gave me: "Integrate by parts ...