All Questions
Tagged with summation real-analysis
1,079
questions
0
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67
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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 ...
- 1,418
2
votes
0
answers
95
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 $x$.
$$...
- 840
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1
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74
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How do we know $x$ is fixed in $\sum_{y \in Y}f(x,y)$?
The following result comes from Analysis I by Terence Tao.
Let $X, Y$ be finite sets, and let $f : X \times Y \to \mathbf{R}$ be a function. Then
$$
\sum_{x \in X}\left(\sum_{y \in Y}f(x,y)\right) = \...
- 1,418
0
votes
1
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51
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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 $$\lim(A_1+A_2+A_3+\dots)=\...
3
votes
0
answers
48
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How to solve $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$? [duplicate]
This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to ...
- 6,332
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0
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43
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Is this a sufficient condition to interchange infinite sums?
I came across this wikipedia article, which has the following result:
Theorem 5: If $a_{m,n}$ is a sequence and $\lim_{n \to \infty} a_{m,n}$ exists uniformly in $m$, and $\lim_{m \to \infty} a_{m,n}$...
- 1
4
votes
1
answer
91
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Why $\infty=\sum_{i=1}^\infty \frac{1}{n+i}\neq\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$?
I was wondering why $\sum_{i=1}^\infty \frac{1}{n+i}$ diverges but $\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$. While assuming integral as limit of series, we find out that:
$$
\int_1^...
- 43
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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
$$
\frac{\partial}{\partial\xi_i}\left[\sum_{i=1}^k\sum_{j=1}^k a_{ij}(\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j)\right] = -2\sum_{...
- 566
1
vote
1
answer
44
views
Help me calculate the triple summation
Problem
We consider
$$
\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 \...
- 566
1
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0
answers
38
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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?
$$
\begin{align}K_m(x)&:=1+\frac{2}{m}\sum_{j=1}^{m-1}(m-j)\cos(jx)
\\&= \frac1m\sum_{j=-(m-1)}^{m-1} (...
- 21
0
votes
2
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54
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what can I deduce from $\sum_{i=1}^n(x_i + y_i) = 0$?
If $x_i$ and $y_i$ are integers. And if I know that $\sum_{i}^{n} x_i = \sum_{i}^{n} y_i = 0$ and that $\sum_{i}^{n}(x_i + y_i) = 0$ what is the best I can deduce about $x_i$ and $y_i$?
Does this ...
user965463
3
votes
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answers
26
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What are these infinite sums of powers of integers, $n^p$, multiplying a quadratic in the Bessel function $J_n(nx)$ and its derivative $J'_n(nx)$?
What are explicit elementary functions of real $x$, for $0 < x < 1$, if they exist, for $p=1$ and $p=3$ of
$$\sum_{n=1}^\infty n^p [J_n(nx)]^2$$
$$\sum_{n=1}^\infty n^{p+1}J_n(nx)J'_n(nx)$$
$$\...
- 31
2
votes
4
answers
156
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A problem on finding the limit of the sum
$$u_{n} = \frac{1}{1\cdot n} + \frac{1}{2\cdot(n-1)} + \frac{1}{3\cdot(n-2)} + \dots + \frac{1}{n\cdot1}.$$
Show that, $\lim_{n\rightarrow\infty} u_n = 0$.
The only approach I can see is either ...
0
votes
1
answer
94
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Showing $\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right ) = \frac{\pi^2}{8}$
Show that
$$\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right) = \frac{\pi^2}{8}$$
I liked this problem because the result is a final answer, and ...
- 1,362
3
votes
2
answers
117
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Why does this proof work: $\sum\limits_{n=1}^ \infty \left(\frac{1}{4n-1} - \frac{1}{4n}\right)= \frac{\ln(64)- \pi}{8}$?
$$f(x):= \sum_{n=1}^ \infty \left(\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}\right)$$
$$f'(x) = \sum_{n=1}^ \infty ( x^{4n-2}- x^{4n-1})= \frac{x^2}{(1+x)(1+x^2)}$$
$$\int_0 ^1 \frac{x^2}{(1+x)(1+x^2)}=...
- 6,332