All Questions
Tagged with real-analysis summation
178
questions with no upvoted or accepted answers
7
votes
0
answers
506
views
A method for evaluating sums/discrete functions by assuming they can be made continuous and differentiable?
Suppose I had a function that satisfied the property $f(x)=f(x-1)+g(x)$. For any $x\in\mathbb N$, it is easy enough to see that this boils down to the statement
$$f(x)=f(0)+\sum_{k=1}^xg(k)$$
If we ...
6
votes
0
answers
104
views
A curious limit: $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}\left[\left(\frac{n}{n+1-i}\right)\right]^{a}f(i) = c\sum\limits_{i\geq 1}f(i)$
I am trying to prove, for the general case whereby $\zeta(\cdot\,,\cdot)$ is the Hurwitz-Zeta function, and $a\in \mathbb{N}$, that
$$\mathcal{L} = \lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\...
6
votes
0
answers
535
views
Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$
Hi I am trying to integrate and obtain a closed form result for
$$
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.
$$
Here is what I tried (but I do not think this is ...
- 9,966
5
votes
0
answers
224
views
Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Now asked on MO here
This paper discusses how to prove that $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $. The first proof on this paper is Euler's original proof:
$$\frac{\sin(\sqrt x)}...
- 6,620
5
votes
0
answers
110
views
Closed form for $\sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$
I need a closed form for $$ \sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$$
where $x\in[1,\infty)$
For $x=1$ we have the sum as $$ \sum_{n=1}^\infty e^{-2\pi n}=\frac{1}{e^{2\pi}-1}$$
For $1<x&...
- 910
5
votes
0
answers
157
views
What is the sum of the reciprocal of the hypotenuse of Pythagorean triangles?
A primitive Pythagorean triplet is a triplet $a^2 + b^2 = c^2$ be where $a,b,c$ have no common factors and is generated by $a = x^2 - y^2, b = 2xy, c = x^2 + y^2$ where $x > y, \gcd(x,y) = 1$. My ...
- 20.8k
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
0
answers
87
views
How deduce $\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ from $\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$?
I know that $\displaystyle\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ for any integer $n \geq 1$ is true.
Now, suppose that $n$ is odd, how show
$$
\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\...
- 323
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
4
votes
0
answers
133
views
Convergence of Variable Base Power Tower $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\ldots}}}$
I’m curious if there is a heuristic method that can be used to solve what appears to be an elementary power tower where every increasing power is decreasing by a half integer. Numerical methods ...
- 127
4
votes
0
answers
97
views
Convergence rate to the average value for a sequence that is uniformly distributed modulo $1$
Let $x_n$ be a sequence of real numbers that asymptotically follow a uniform distribution modulo $1$ (for example, $x_n=n\sqrt{2}$ with $n$ positive integer $\leq N$ and $N\rightarrow \infty$). It is ...
- 17.1k
4
votes
0
answers
376
views
Can we write $\ln(x) $ as an infinite sum of $n$ th roots?
Is there a real sequence $0 \leq a_n $ such that for $x > 1 $ we have :
$$\ln(x) = a_0 + \sum_{n=1}^{\infty} (-1)^n \cdot a_n \cdot x^{1/n}$$
Or
$$\ln(x) = a_0 + \sum_{n=1}^{\infty} (-1)^{1+n} ...
- 16.4k
4
votes
0
answers
427
views
Convergence of sum involving Bessel function of the first kind
I am interested in finding any upper bound for the following series:
$$\displaystyle f(x) := \sum_{n \in \mathbb{Z}^d \backslash \{0\}} |n|^{-d/2}J_{d/2}(k|n|)e^{inx},$$
where $d \geqslant 2$, $x \...
- 1,145
4
votes
0
answers
106
views
Arriving at $1+2+3+ \cdots = -1/12$ by Trying to Solve the Basel Problem?
I know there are probably many questions about the sum $1+2+3+4+\cdots = -1/12$, but I am wondering about the method I used which lead me to this result which was not my intention.
I am lacking ...
- 677
4
votes
1
answer
141
views
Interchange summation and differentiation for ONB
Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$
Now assume we have that $$\frac{d}{dx}e_n = \lambda_n e_n.$$
Assume $f \in H^1[0,1],$ so i.e. $||f'||_{L^2} < \infty$
...
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