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Results tagged with sequences-and-series
Search options answers only
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user 11919
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
8
votes
Does $\sum_n \frac{\sin n}n$ converge absolutely?
No, it doesn't. It is easy to see that for any positive integer $m$ we have
$$ \left|\frac{\sin(m)}{m}\right| + \left|\frac{\sin(m+1)}{m+1}\right| > \frac{1}{6m}.$$
Summing this up over all even posi …
- 105k
10
votes
Accepted
About $\lim_{n\to +\infty} n\prod_{k=1}^{n-1}\cos^2(k)$
It follows from the irrationality of $\pi$ and Weyl's criterion that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the integers $k\in\{1,\dotsc,n-1 …
- 105k
24
votes
Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?
The affirmative answer follows from my response to this related question.
EDIT. Noam Elkies gave a nicer and more general argument here.
- 105k
23
votes
Accepted
Evaluating an infinite sum related to $\sinh$
Here is a proof, surely not the simplest. Using
$$\frac1{\sinh(n\pi)}=\frac{2e^{-\pi n}}{1-e^{-2\pi n}}=2\sum_{m\text{ odd}}e^{-mn\pi}$$
and
$$2\sum_{n\text{ odd}}\frac{x^n}{n}=-2\ln(1-x)+\ln(1-x^2 …
- 105k
16
votes
Series whose convergence is not known
Assume that $A\subset\mathbb{N}$ contains no 3-term arithmetic progression. It was conjectured by Erdős and Turán that $\sum_{a\in A}\frac{1}{a}$ converges. As far as I know this is open, although we …
- 105k
2
votes
Accepted
convergence for a series
This is not a research level question, but I feel like answering it. Simply, use the elementary estimate
$$\log(m!)\sim m\log m.$$
The symbol $\sim$ means that the ratio of the two sides tends to $1$. …
- 105k
12
votes
Is this formula expressing $\operatorname{erf} \left(\tfrac 1 {\sqrt 2}\right)$ as an infini...
WolframAlpha returns this result promptly, so it is probably known or straightforward to show. See here.
- 105k
3
votes
When is there a closed form for $\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$?
Closed form is a subjective term. At any rate, our understanding of these sums is quite limited, e.g. it is not known if $\sum_{n=1}^\infty n^{-5}$ is an irrational number (same for any smaller negati …
- 105k
7
votes
Proof of equidistribution theorem for exponential coefficients
For the coefficients $2^n$ the equidistribution theorem fails. In fact it is easy to exhibit an irrational $a$ such that the sequence $(2^na)_{n=0}^\infty$ is not even dense in $(0,1)$ modulo $1$. For …
- 105k
10
votes
Accepted
Growth of a linear recurrent sequence
First note that $\frac{1+i\sqrt{2}}{1-i\sqrt{2}}\in\mathbb{Q}(\sqrt{-2})$ is not a root of unity, because it does not equal $\pm 1$. Therefore Baker's famous theorem shows that, for some effectively c …
- 105k
1
vote
Accepted
Monotonic sequence (edited)
I guess the OP means that the values are ordered linearly, i.e. $a_1<\dots<a_m$. In that case the statement is even true for $s=mn-n+2$.
To see this, define the numbers $1\leq c(i,k)\leq m$ such tha …
- 105k
8
votes
Accepted
Proving convergence of sum over $\mathbb{Z}^n$
The sum diverges already for $n=4$. To see this, let $L\geq K$ be a dyadic parameter, and consider the contribution of $q_1\asymp L^{1/8}$, $q_2\asymp L^{1/6}$, $q_3\asymp L^{1/4}$, $q_4\asymp L^{1/2} …
- 105k
18
votes
Accepted
An inequality involving square roots and sums
For every $n\in\{1,\dotsc,N\}$, we have
$$2\sqrt{\sum_{i\leq n} a_i}-2\sqrt{\sum_{i\leq n-1} a_i}=\frac{2a_n}{\sqrt{\sum_{i\leq n} a_i}+\sqrt{\sum_{i\leq n-1} a_i}}>\frac{a_n}{\sqrt{\sum_{i\leq n} a_i …
- 105k
9
votes
Accepted
Is it possible to determine whether the sequence $\,a_0=p,\;a_{n+1}=(a_n-2)\cdot a_n+2\,$ wi...
Your question can be reformulated as follows.
Question. If $p$ is a prime, does there always exist a positive integer $n$ such that $(p-1)^{2^n}+1$ is also a prime?
I believe that this question is o …
- 105k
9
votes
Accepted
Determining the asymptotic behavior of a series
This is my third response. I claim that for $0 < t < 1$ we have the uniform bounds
$$ \liminf_{n\to\infty}\ nt f_n(t) = \frac{-1}{\log\mu}+O(1),$$
$$ \limsup_{n\to\infty}\ nt f_n(t) = \frac{-1}{\log\m …