All Questions
239
questions
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
0
votes
2
answers
90
views
Isn't my book using the summation notation incorrectly when writing $\lim_{\Delta x\to0}\sum_{n=1}^{N}f(x)\Delta x$?
My book was introducing the concept of integrals and wrote this:
$$\text{Area under the curve of $f(x)$}=\lim_{\Delta x\to0}\sum_{n=1}^{N}f(x)\Delta x\tag{1}$$
My problem with $(1)$ is that there is ...
- 3,186
5
votes
1
answer
78
views
Find the minimum posssible integer value of the summation
Let $f(x)$ is a continuous, increasing and positive value function in the interval $[0,a]$ such that
$$\int_0^af(x)dx=20$$
Then find the minimum posssible integer value of the following summation
$$a\...
- 506
2
votes
1
answer
175
views
Infinite summation of recursive integral
Let $I_n=\int_{0}^{1}e^{-y}y^n\ dy$, where $n$ is non-negative integer. Find $\sum_{n=1}^{\infty}\frac{I_n}{n!}.$
I first solved $I_n$ and obtained $$I_n=-\frac{1}{e}+nI_{n-1} \\
\hspace{35mm} =-\...
- 175
6
votes
1
answer
215
views
Ramanujan's q function
I stuck at the following problem:
Let
\begin{equation}
Q(n) := \sum_{k \geq 0}\frac{(n-1)_k}{n^k}
\end{equation}
where $(n)_k = n (n-1) \ldots (n-k + 1)$.
I want to show the following equation:
\begin{...
- 1,060
6
votes
3
answers
325
views
Integral Representation of a Double Sum
Let us assume we know the value of $x$ and $y$. I'm trying to write the following double sum as an integral. I went through many pages and saw various methods but I'm completely lost with my problem.
$...
user977415
9
votes
2
answers
534
views
Integral of the shark function
Messing around with functions is my hobby, I am asking this for fun, and maybe as a little challenge.
I gave this style of function the name "Shark function" because it looks like the shark'...
0
votes
1
answer
139
views
Evaluate $\sum^{\infty}_{n=1} \frac{1}{n{2n\choose n}}$ [duplicate]
I was having trouble with the sum
$$\sum^{\infty}_{n=1} \frac{1}{n {2n\choose n} }$$
My Attempt
$$S=\sum^{\infty}_{n=1} \frac{1}{n {2n\choose n} }=\sum^{\infty}_{n=1} \frac{n!\; n!}{n\;(2n)! }=\sum^{\...
- 1,373
3
votes
1
answer
276
views
Find limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$.
Find the following limit:
$$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$$
I had an idea of using upper Riemann sum for function $x^4$ on interval $[0,1]$ but I don't know how to deal with $...
- 41
2
votes
0
answers
70
views
Solving $\displaystyle I = \int_{0}^{1} \left (( \ln u)^{-z} - \sum_{k=0}^{n-1} \frac {(\ln u)^{k-z}}{u \cdot k!} \right ) \mathrm {d}u$ with $z>0$.
$$\displaystyle I = \int_{0}^{1} \left (( \ln u)^{-z} - \sum_{k=0}^{n-1} \frac {(\ln u)^{k-z}}{u \cdot k!} \right ) \mathrm {d}u$$
My attempt :
Distributing the integration over the two parts as ...
- 955
1
vote
2
answers
201
views
Integration and summation: prove $399< \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{40000}}<400$ is false. [duplicate]
I need help proving that the following statement is false:
$$
399 < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}}
+ \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{40000}} < 400.
$$
I tried to bound ...
- 1,237