All Questions
101
questions
-1
votes
1
answer
70
views
How to give this sum a bound?
Let $x,y\in\mathbb{Z},$ consider the sum below
$$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$
is there anything I could do to give this sum a ...
- 329
0
votes
0
answers
45
views
Is this a sufficient condition to interchange infinite sums?
I came across this wikipedia article, which has the following result:
Theorem 5: If $a_{m,n}$ is a sequence and $\lim_{n \to \infty} a_{m,n}$ exists uniformly in $m$, and $\lim_{m \to \infty} a_{m,n}$...
- 31
0
votes
0
answers
22
views
Please help me with the partial differentiation of a matrix elementwise
Background
Help me calculate the triple summation
Problem
We want to show that
$$
\frac{\partial}{\partial\xi_i}\left[\sum_{i=1}^k\sum_{j=1}^k a_{ij}(\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j)\right] = -2\sum_{...
- 590
1
vote
1
answer
45
views
Help me calculate the triple summation
Problem
We consider
$$
\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\xi_i)(x_{j\nu}-\xi_j) = \sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\bar{x}_i)(x_{j\nu}-\bar{x}_j) - 2n \...
- 590
0
votes
2
answers
54
views
what can I deduce from $\sum_{i=1}^n(x_i + y_i) = 0$?
If $x_i$ and $y_i$ are integers. And if I know that $\sum_{i}^{n} x_i = \sum_{i}^{n} y_i = 0$ and that $\sum_{i}^{n}(x_i + y_i) = 0$ what is the best I can deduce about $x_i$ and $y_i$?
Does this ...
user965463
3
votes
1
answer
63
views
Proving Existence of Sequence and Series Satisfying Hardy-Littlewood Convergence Condition Prove that for every $\vartheta, 0 < \vartheta < 1$,
Prove that for every $\vartheta, 0 < \vartheta < 1$, there exists a sequence $\lambda_{n}$ of positive integers and a series $\sum_{n=1}^{\infty} a_{n}$ such that:
\begin{align*}
(i) & \...
0
votes
2
answers
62
views
Comparing integral with a sum
Show that \begin{equation}\sum_{m=1}^k\frac{1}{m}>\log k.\end{equation}
My intuition here is that the LHS looks a lot like $\int_1^k\frac{1}{x}\textrm{d}x$, and this evaluates to $\log k$. To ...
- 135
3
votes
1
answer
85
views
Proving two averages are asymptotically equivalent
Suppose $f(n)\sim g(n)$ as $n\to\infty$. Is it necessarily true that \begin{equation}\frac{1}{n}\sum_{k=1}^n|f(k+1)-f(k)|\sim\frac{1}{n}\sum_{k=1}^n|g(k+1)-g(k)|\end{equation}
as $n\to\infty$?
...
- 31
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:
$$\frac{\sin(\sqrt x)}...
- 6,620
-1
votes
3
answers
170
views
How do you find the value of $\sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{n^2}$? [closed]
Extra information which may be useful is that $\sum_{n=1}^{\infty} \frac{1}{n^2}$ equals $\frac{\pi^2}{6}$ (Euler's solution to the Basel Problem).
1
vote
0
answers
50
views
Does this series converge ? $\sum_{n=1}^{\infty} (-1)^n(\sqrt[n]{a}-1), a \geq 1$ Hint: $\lim_{n \to \infty} \frac{\sqrt[n]{a}-1}{1/n} = \ln{a}$
Does this series converge ?
$$\sum_{n=1}^{\infty} (-1)^n(\sqrt[n]{a}-1), a \geq 1$$
Hint: use the fact that
$$\lim_{n \to \infty} \frac{\sqrt[n]{a}-1}{1/n} = \lim_{h \to 0} \frac{a^h-1}{h} = \ln{a}$$
...
- 1,135
1
vote
1
answer
86
views
Find radius of convergence of the power serie $\sum_{n=1}^{\infty} (\sqrt{n} - 1)^{\sqrt{n}}z^n$
Find radius of convergence of the power serie $\sum_{n=1}^{\infty} (\sqrt{n} - 1)^{\sqrt{n}}z^n$
I first tried to use the Root Test.
$L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \...
- 1,135
0
votes
2
answers
69
views
Find radius of convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n(2z)^n}{n}$
Sorry for my last duplicate question. But for this question here, I did not find the same question
Find radius of convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n(2z)^n}{n}$
$L = \lim_{n \to \infty} |\...
- 1,135
0
votes
0
answers
14
views
Find radius of convergence of $\sum_{n=1}^{\infty}\frac{n!}{n^n}z^n$ [duplicate]
Find radius of convergence of $\sum_{n=1}^{\infty}\frac{n!}{n^n}z^n$
My attempt:
$L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$ with $a_n = n!/n^n$. This gives
$L = \lim_{n \to \infty} |(n+1)\frac{n^...
- 1,135
1
vote
0
answers
56
views
Prove for a sequence ($a_n$)$_n$ with converging partial sum $s_n = \sum_{k=1}^{n}a_k$ it holds that for a bounded, monotonically decreasing sequence
Prove that for a sequence ($a_n$)$_n$
with converging partial sum
$$s_n = \sum_{k=1}^{n}a_k$$ , it holds that for a bounded, monotonically decreasing sequence ($c_n$)$_n$, the series
$$\sum_{n=1}^{\...
- 1,135