All Questions
Tagged with real-analysis summation
1,083
questions
2
votes
1
answer
29
views
If a sequence $a_n$ satisfies the following two properties, does $\sum_{k=1}^{\infty} (\sum_{n=1}^{\infty} \frac{1}{a_n^k} - L)$ converge?
Let $a_n$ be a positive, increasing sequence satisfying the following two properties:
$S_k :=\displaystyle\sum_{n=1}^{\infty} \frac{1}{a_n^k}$ converges for all $k \in \mathbb{N}$.
And $\displaystyle\...
- 340
-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
1
vote
1
answer
29
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Subset of index that minimizes a sum of real values
Given a series of real numbers $c_1, \ldots, c_n$ with $ n \in \mathbb{N} $, is there an algorithm or method to find the subset of indices such that the absolute value of the sum of the values within ...
- 1,070
3
votes
0
answers
162
views
Closed form for $\sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$?
I've found this sum:
$$S(n) = \sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$$
The inner sum is elliptical iirc, but perhaps the double sum has a nice expression. We can ...
- 1,940
0
votes
1
answer
68
views
Showing $\sum_{y \in Y} f(x_0, y) = \sum_{(x,y) \in \{x_0\} \times Y} f(x, y)$.
Let $X, Y$ be finite sets, and let $f: X \times Y \to \mathbb{R}$ be a function. I am trying to show that
$$
\sum_{y \in Y} f(x_0, y) = \sum_{(x,y) \in \{x_0\} \times Y} f(x, y)
$$
by using the ...
- 1,454
2
votes
0
answers
97
views
Fractional part of a sum
Define for $n\in\mathbb{N}$ $$S_n=\sum_{r=0}^{n}\binom{n}{r}^2\left(\sum_{k=1}^{n+r}\frac{1}{k^5}\right)$$
I need to find $\{S_n\}$ for $n$ large where $\{x\}$ denotes the fractional part of $x$.
$$...
- 910
0
votes
1
answer
74
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How do we know $x$ is fixed in $\sum_{y \in Y}f(x,y)$?
The following result comes from Analysis I by Terence Tao.
Let $X, Y$ be finite sets, and let $f : X \times Y \to \mathbf{R}$ be a function. Then
$$
\sum_{x \in X}\left(\sum_{y \in Y}f(x,y)\right) = \...
- 1,454
0
votes
1
answer
52
views
How to prove the sum of limits theorem for a finite N number of limits? [duplicate]
I was reading a book with sequences and it proved that given two sequences $A$ and $B$ which both converge, then $\lim(A+B) =\lim(A)+\lim(B)$.
However, the sum of $N$ limits $$\lim(A_1+A_2+A_3+\dots)=\...
3
votes
0
answers
48
views
How to solve $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$? [duplicate]
This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to ...
- 6,620
0
votes
0
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45
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Is this a sufficient condition to interchange infinite sums?
I came across this wikipedia article, which has the following result:
Theorem 5: If $a_{m,n}$ is a sequence and $\lim_{n \to \infty} a_{m,n}$ exists uniformly in $m$, and $\lim_{m \to \infty} a_{m,n}$...
- 31
4
votes
1
answer
91
views
Why $\infty=\sum_{i=1}^\infty \frac{1}{n+i}\neq\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$?
I was wondering why $\sum_{i=1}^\infty \frac{1}{n+i}$ diverges but $\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$. While assuming integral as limit of series, we find out that:
$$
\int_1^...
- 43
0
votes
0
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22
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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
1
vote
0
answers
38
views
When I was looking at the proof of Fejer's theorem, I encountered a problem in the derivation of a formula. [closed]
How did we get the last equation? Why can the summation be converted into a square term?
$$
\begin{align}K_m(x)&:=1+\frac{2}{m}\sum_{j=1}^{m-1}(m-j)\cos(jx)
\\&= \frac1m\sum_{j=-(m-1)}^{m-1} (...
- 21
0
votes
2
answers
54
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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 ...
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