All Questions
11
questions
0
votes
1
answer
41
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How to express $\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $ in terms of an integral?
I have this sum
$$\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $$
where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. I want to transform summation to an integral ...
- 51
0
votes
0
answers
33
views
Question on transforming a sum to an integral using the Euler–Maclaurin formula.
I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum
$$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+...
- 51
2
votes
1
answer
248
views
What are the conditions for Ramanujan's Master Theorem to hold?
Ramanujan's Master Theorem states that if
$$f(x) = \sum_{k=0}^{\infty} \frac{\phi(k)}{k!}(-x)^k$$
then $$\int_{0}^{\infty}x^{s-1}f(x)\ dx = \Gamma(s)\phi(-s).$$
But there are obviously some conditions ...
- 359
6
votes
3
answers
325
views
Integral Representation of a Double Sum
Let us assume we know the value of $x$ and $y$. I'm trying to write the following double sum as an integral. I went through many pages and saw various methods but I'm completely lost with my problem.
$...
user977415
2
votes
4
answers
94
views
Why is the inequality $\sum_{n=1}^{\infty} \frac{1}{n^2} \leq 1 + \int_1^{\infty} \frac{1}{x^2}$ true?
$$\sum_{n=1}^{\infty} \frac{1}{n^2} \leq 1 + \int_1^{\infty} \frac{1}{x^2}dx$$
I'm having trouble figuring out why the inequality above is true. I understand the following inequality:
$$\int_1^{\...
- 461
4
votes
2
answers
299
views
Transfer between integrals and infinite sums
So I was watching a video on YouTube about how $$\sum_{i=1}^\infty \frac{\chi(i)}{i} = \frac{\pi}{4}$$ (note that $\chi(i) = 0$ for even numbers $i$, $1$ for $\text{mod}(i, 4) = 1$, and $-1$ for $\...
- 2,357
0
votes
1
answer
55
views
Find the primitive function of $\sum_{n=1}^{\infty}\frac{(-1)^n(2n+2)}{n!}(x-1)^{2n+1}$
Find the primitive function of
$\sum_{n=1}^{\infty}\frac{(-1)^n(2n+2)}{n!}(x-1)^{2n+1}$
My attempt:
In order to integrate, I'm trying to find the radius of convergence:
Let $t=(x-1) \Rightarrow \...
- 507
0
votes
1
answer
1k
views
How to find sum of the power series $\sum_{n=1}^{\infty} x^2 e^{-nx}$?
I got this power series $$\sum_{n=1}^{\infty} x^2 e^{-nx} $$
And i need to prove uniform convergence when $x \in[0;1]$ and find this sum. I have proven uniform convergence, but i have no idea how to ...
- 535
1
vote
2
answers
217
views
Sophomore's Dream : integral not defined in x=0
Sophomore's dream is the identity that states
\begin{equation}
\int_0^1 x^x dx = \sum\limits_{n=1}^\infty (-1)^{n+1}n^{-n}
\end{equation}
The proof is found using the series expansion for $e^{-x\...
- 11
1
vote
2
answers
2k
views
Power series and shifting index
First I have to find the power series represantion for the following function:
$$\ f(x) = \ln(1+x)$$
I tried the following:
$$\ \frac{d}{dx}\Big(\ln(1+x)\Big) = \frac{1}{1+x} = \sum_{n=0}^\infty(-...
- 215
1
vote
1
answer
1k
views
Integral of Summation (power series)
Could someone guide me through this process, I am confused on how you can take an integral of the factorial or whatever is going on in the problem.
In the context of this problem, the summation is ...
- 927