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
views
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 $x$.
$$...
- 910
0
votes
1
answer
74
views
How do we know $x$ is fixed in $\sum_{y \in Y}f(x,y)$?
The following result comes from Analysis I by Terence Tao.
Let $X, Y$ be finite sets, and let $f : X \times Y \to \mathbf{R}$ be a function. Then
$$
\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 $$\lim(A_1+A_2+A_3+\dots)=\...
3
votes
0
answers
48
views
How to solve $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$? [duplicate]
This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to ...
- 6,620
0
votes
0
answers
45
views
Is this a sufficient condition to interchange infinite sums?
I came across this wikipedia article, which has the following result:
Theorem 5: If $a_{m,n}$ is a sequence and $\lim_{n \to \infty} a_{m,n}$ exists uniformly in $m$, and $\lim_{m \to \infty} a_{m,n}$...
- 31
4
votes
1
answer
91
views
Why $\infty=\sum_{i=1}^\infty \frac{1}{n+i}\neq\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$?
I was wondering why $\sum_{i=1}^\infty \frac{1}{n+i}$ diverges but $\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$. While assuming integral as limit of series, we find out that:
$$
\int_1^...
- 43
0
votes
0
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
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?
$$
\begin{align}K_m(x)&:=1+\frac{2}{m}\sum_{j=1}^{m-1}(m-j)\cos(jx)
\\&= \frac1m\sum_{j=-(m-1)}^{m-1} (...
- 21
0
votes
2
answers
54
views
what can I deduce from $\sum_{i=1}^n(x_i + y_i) = 0$?
If $x_i$ and $y_i$ are integers. And if I know that $\sum_{i}^{n} x_i = \sum_{i}^{n} y_i = 0$ and that $\sum_{i}^{n}(x_i + y_i) = 0$ what is the best I can deduce about $x_i$ and $y_i$?
Does this ...
user965463
3
votes
0
answers
28
views
What are these infinite sums of powers of integers, $n^p$, multiplying a quadratic in the Bessel function $J_n(nx)$ and its derivative $J'_n(nx)$?
What are explicit elementary functions of real $x$, for $0 < x < 1$, if they exist, for $p=1$ and $p=3$ of
$$\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
158
views
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
95
views
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 ...
- 1,433
3
votes
2
answers
117
views
Why does this proof work: $\sum\limits_{n=1}^ \infty \left(\frac{1}{4n-1} - \frac{1}{4n}\right)= \frac{\ln(64)- \pi}{8}$?
$$f(x):= \sum_{n=1}^ \infty \left(\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}\right)$$
$$f'(x) = \sum_{n=1}^ \infty ( x^{4n-2}- x^{4n-1})= \frac{x^2}{(1+x)(1+x^2)}$$
$$\int_0 ^1 \frac{x^2}{(1+x)(1+x^2)}=...
- 6,620
0
votes
4
answers
196
views
How to evaluate $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$
I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice.
$$\sum_{n=3}^ \...
- 6,620
5
votes
1
answer
183
views
How to rigorously prove that $\sum\limits_{n=1}^ \infty( \frac{1}{4n-1} - \frac{1}{4n} )=\frac{\ln(64)- \pi}{8}$?
How to rigorously prove that $\sum\limits_{n=1}^ \infty\left( \frac{1}{4n-1} - \frac{1}{4n}\right) =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \left(\frac{x^{4n-1}}{4n-1} - \frac{x^...
- 6,620
3
votes
1
answer
63
views
Proving Existence of Sequence and Series Satisfying Hardy-Littlewood Convergence Condition Prove that for every $\vartheta, 0 < \vartheta < 1$,
Prove that for every $\vartheta, 0 < \vartheta < 1$, there exists a sequence $\lambda_{n}$ of positive integers and a series $\sum_{n=1}^{\infty} a_{n}$ such that:
\begin{align*}
(i) & \...
4
votes
1
answer
169
views
Prove that sum of integrals $= n$ for argument $n \in \mathbb{N}_{>1}$
ORIGINAL QUESTION (UPDATED):
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \...
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
0
votes
0
answers
21
views
Changing the order of the sum in a measure question [duplicate]
Let $(X,\mathcal{A})$ be a measurable space and $(\mu_n)$ be a sequence of measures which satisfies $\mu_n(X)=1$. I want to show that the function defined as
$$
\nu(E)=\sum_{n=1}^{\infty}2^{-n}\mu_n(E)...
- 53
6
votes
1
answer
229
views
Is it always true that if $x_n\to0$, $y_n\to0$ there exist $\epsilon_n\in\{-1,1\}$ such that both $\sum\epsilon_nx_n$and $\sum\epsilon_ny_n$ converge? [duplicate]
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \...
- 6,620
2
votes
1
answer
111
views
For $a>1$ and a fixed $N$ does $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \left(1- \frac{a}{a+N +k} \right)$ converge?
I was trying to find a proof for Raabe's Ratio test: If $x_n$ is a positive sequence of real numbers, if $\lim\limits_{n \to \infty } n \left(\frac{x_n}{x_{n+1}}-1 \right) >1$ then $\sum\limits_{...
- 6,620
2
votes
1
answer
87
views
Convergence of the series $f(a_n)a_n$
Let $f : \mathbb{R} \to \mathbb{R}$ a monoton function in $[−r, r]$ for some $r>0$. Prove that the if $$\sum a_n$$ converge absolutely then $$\sum f(a_n)a_n$$ converges absolutely
I have not idea ...
- 145
1
vote
1
answer
123
views
Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
1
vote
0
answers
30
views
Insufficient boundedness for integration via Darboux sums
I have the simple integral $$\int_0^1 \frac{dx}{\sqrt{1-x}},$$ which I would like to constrain between two Darboux sums whose summands have absolute values not all zero. My attempt in doing so begins ...
- 76
6
votes
2
answers
502
views
How to perform this sum
I encountered this sum
$$S(N,j)= \frac{2 \sqrt{2}h(-1)^j}{N+1}\cdot\sum _{n=1}^{\frac{N}{2}}
\frac{\sin ^2\left(\frac{\pi j n}{N+1}\right)}{\sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}},$$
...
- 712
3
votes
1
answer
119
views
Does $\prod\limits_{n=1}^{\infty}\int\limits_{-1}^{1}\left|\frac{\lfloor t^{1-n}\rfloor}{\lfloor t^{-n}\rfloor}\right|dt$ converge to $1$?
This is the first $40$ partial products on Desmos (Desmos gave me "undefined" for everything higher):
Looking at this, it doesn't appear to converge but then, while attempting to get Desmos ...
- 1,688
5
votes
1
answer
179
views
What is $\int\limits_0^1\left\{\frac{1}{x\left\{\frac{1}{x}\right\}}\right\}dx$?
$\{x\}$ is the fractional part of $x$.
$\{x\}=x-\lfloor x\rfloor$
I ended up with this double summation:
$$\lim\limits_{\substack{a\to\infty\\b\to\infty}}\sum_{m=1}^{a}\sum_{n=0}^{b}\left(\frac{1}{m}\...
- 1,688
-3
votes
2
answers
155
views
Prove : $a^2+b^2+c^2≤ \frac{2}{3}\cdot (a^3+b^3+c^3)+1$ [closed]
Prove that by Cauchy inequality:
$a^2+b^2+c^2≤ \frac{2}{3}\cdot (a^3+b^3+c^3)+1$
$a,b,c$ are a positive numbers
$a*b*c=1$
- 33
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
0
votes
0
answers
31
views
Can we compare the arithmetic mean of the ratios given the comparison between individual arithmetic means?
I have positive real random numbers $u_1,\ldots,u_n$ and $v_1,\ldots,v_n$ and $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$.
I know that the arithmetic mean of $u_i$'s is greater than the arithmetic mean of $...
- 452
2
votes
1
answer
100
views
How to prove that for fixed number $m$ positive numbers the sequence $\frac{\sum\limits_{k=1}^ma_k^{n+1}}{\sum\limits_{k=1}^ m a_k^{n}}$ is monotone?
I saw this question in my book
Let $a_1 , a_2 , \dots,a_m$ be fixed positive numbers and $S_n =\frac{\sum \limits_{k=1}^ m a_k^{n}}{m}$ Prove that $\sqrt[n] {S_n} $ is monotone increasing sequence
...
- 6,620
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:
$$\frac{\sin(\sqrt x)}...
- 6,620
2
votes
0
answers
365
views
A sum of two curious alternating binoharmonic series
Happy New Year 2024 Romania!
Here is a question proposed by Cornel Ioan Valean,
$$\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{2^{2n}}\binom{2n}{n}\sum_{k=1}^n (-1)^{k-1}\frac{H_k}{k}-\sum_{n=1}^{\infty}(-1)...
- 5,495
2
votes
1
answer
220
views
Proving a property related to $M/M/c$ queues - Queueing theory.
My goal is to show that in a $M/M/c$ queueing system it is satisfied that
$$ L_s = L_q + \frac{\lambda}{\mu}, $$
where $L_s$ represents the average number of costumers in the system, $L_q$ represents ...
- 1,141
2
votes
0
answers
63
views
Cases where transcendental numbers can add up to a rational number? [closed]
Other than sums like $π + (1 - π)$, obviously. Can two transcendental numbers add up to a rational number? Or how about an infinite series of them?
- 453
1
vote
1
answer
60
views
Why does $f^{(k)}(0)$ exists in Rudin's PMA Corollary 8.1?
is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
2
votes
4
answers
273
views
How did Rudin change the order of the double sum $\sum_{n=0}^\infty c_n\sum_{m=0}^n\binom nma^{n-m}(x-a)^m$?
I see many people change the order of sum but I don't understand how they did that.
Is there is a way to change the order of the sum, $\sum\limits_{k=a}^n\sum\limits_{j=b}^m X_{j,k}$ and $\sum\...
0
votes
1
answer
142
views
How to rigorously prove that $e^x = \sum\limits_{n=0}^ \infty \frac{x^n}{n!}$ without defining derivatives? [duplicate]
In my problem book, there was a question: By defining $e= \lim\limits_{n \to \infty}\left( 1+\frac{1}{n} \right) ^n$ prove that $e^x = \sum\limits_{n=0}^ \infty \frac{x^n}{n!}$. this is a strange ...
- 6,620
0
votes
0
answers
64
views
Converting complex-exponential summation to Fresnel integrals
I have a summation
$$S = \sum_{n=0}^N e^{-jn^2a}, \ a\ne 0, \ n\in\{0,1,\cdots,N\}$$
and it can be approximated by
$$S\approx I = \int_{n=0}^N e^{-jn^2a}dn$$
when $N$ is sufficiently large. ...
- 71
3
votes
1
answer
145
views
Do the normal numbers form a Borel set?
Normal numbers have a 'random' expansion. For example, in base 10 it means that all digits $0,1,\dots,9$ occur 'equally often' in its decimal expansion. A longstanding open problem is: is $\pi$ a ...
- 727
1
vote
1
answer
91
views
Evaluating a finite sum.
Amid one exercise I was solving, I came across the following finite sum:
$$ \sum_{n=0}^{N} n\left(\frac{3}{2}\right)^n.$$
This sum was evaluated in one of my classes, but I don't understand/agree with ...
- 1,141
0
votes
1
answer
69
views
Taking constant outside the summation
In the below picture they have taken $a$ outside? Is it allowed? Is both terms involving summation are equal?
My attempt
Term on left hand side is equal to $|\frac{1}{n}(a_1+... +a_n)-a|$
Whereas the ...
3
votes
0
answers
63
views
Is there any function in which the Maclaurin series evaluates to having prime numbered powers and factorials? [duplicate]
I am searching for any information or analysis regarding the functions
$$f(x)=\sum_{n=1}^{\infty}\frac{x^{p\left(n\right)}}{\left(p\left(n\right)\right)!}$$
or
$$g(x)=\sum_{n=1}^{\infty}\frac{\left(-1\...
- 41
3
votes
2
answers
296
views
if $\lim\limits_{n \to \infty} b_n =0 $ then how to prove that $\lim\limits_{n \to \infty} \sum\limits_{k =1} ^n \frac{b_k}{n+1-k}=0$
in Problems in Mathematical Analysis I problem 2.3.16 a),
if $\lim\limits_{n \to \infty}a_n =a$, then find $\lim\limits_{n \to \infty} \sum\limits_{k=1 }^n \frac{a_k}{(n+1-k)(n+2-k)}$
The proof that ...
- 6,620