All Questions
Tagged with real-analysis summation
1,083
questions
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
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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:
$$\lim\limits_{\substack{a\to\infty\\b\to\infty}}\sum_{m=1}^{a}\sum_{n=0}^{b}\left(\frac{1}{m}\...
- 1,688
-3
votes
2
answers
155
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Prove : $a^2+b^2+c^2≤ \frac{2}{3}\cdot (a^3+b^3+c^3)+1$ [closed]
Prove that by Cauchy inequality:
$a^2+b^2+c^2≤ \frac{2}{3}\cdot (a^3+b^3+c^3)+1$
$a,b,c$ are a positive numbers
$a*b*c=1$
- 33
3
votes
1
answer
85
views
Proving two averages are asymptotically equivalent
Suppose $f(n)\sim g(n)$ as $n\to\infty$. Is it necessarily true that \begin{equation}\frac{1}{n}\sum_{k=1}^n|f(k+1)-f(k)|\sim\frac{1}{n}\sum_{k=1}^n|g(k+1)-g(k)|\end{equation}
as $n\to\infty$?
...
- 31
0
votes
0
answers
31
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Can we compare the arithmetic mean of the ratios given the comparison between individual arithmetic means?
I have positive real random numbers $u_1,\ldots,u_n$ and $v_1,\ldots,v_n$ and $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$.
I know that the arithmetic mean of $u_i$'s is greater than the arithmetic mean of $...
- 452
2
votes
1
answer
100
views
How to prove that for fixed number $m$ positive numbers the sequence $\frac{\sum\limits_{k=1}^ma_k^{n+1}}{\sum\limits_{k=1}^ m a_k^{n}}$ is monotone?
I saw this question in my book
Let $a_1 , a_2 , \dots,a_m$ be fixed positive numbers and $S_n =\frac{\sum \limits_{k=1}^ m a_k^{n}}{m}$ Prove that $\sqrt[n] {S_n} $ is monotone increasing sequence
...
- 6,620
5
votes
0
answers
224
views
Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Now asked on MO here
This paper discusses how to prove that $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $. The first proof on this paper is Euler's original proof:
$$\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}, $$
where $L_s$ represents the average number of costumers in the system, $L_q$ represents ...
- 1,141
2
votes
0
answers
63
views
Cases where transcendental numbers can add up to a rational number? [closed]
Other than sums like $π + (1 - π)$, obviously. Can two transcendental numbers add up to a rational number? Or how about an infinite series of them?
- 453
1
vote
1
answer
60
views
Why does $f^{(k)}(0)$ exists in Rudin's PMA Corollary 8.1?
is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
2
votes
4
answers
273
views
How did Rudin change the order of the double sum $\sum_{n=0}^\infty c_n\sum_{m=0}^n\binom nma^{n-m}(x-a)^m$?
I see many people change the order of sum but I don't understand how they did that.
Is there is a way to change the order of the sum, $\sum\limits_{k=a}^n\sum\limits_{j=b}^m X_{j,k}$ and $\sum\...
0
votes
1
answer
142
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How to rigorously prove that $e^x = \sum\limits_{n=0}^ \infty \frac{x^n}{n!}$ without defining derivatives? [duplicate]
In my problem book, there was a question: By defining $e= \lim\limits_{n \to \infty}\left( 1+\frac{1}{n} \right) ^n$ prove that $e^x = \sum\limits_{n=0}^ \infty \frac{x^n}{n!}$. this is a strange ...
- 6,620
0
votes
0
answers
64
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Converting complex-exponential summation to Fresnel integrals
I have a summation
$$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