Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with sequences-and-series
Search options answers only
not deleted
user 6101
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.
22
votes
Closed form of an infinite series
Denote $c_n:={(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}$ the $n$-th term of the series. We have for all $k\ge0$
$$c_{3k}=0,$$
$$c_{3k+1}= (-1)^{k+1}\ …
- 60.6k
19
votes
Accepted
power series of the reciprocal... does a recursive formula exist for the coefficients
Assume $b_0=1$ to simplify things. You want a closed formula for the recursively defined sequence $$d_0=1$$ $$d_n=-\sum_{k=0}^{n-1}d_kb_{n-k}. $$
Let $\alpha=(\alpha_1,\dots,\alpha_r)\in \mathbb{N}_ …
- 60.6k
17
votes
Accepted
Characterizing positivity of formal group laws
Given $\phi(x)\in\mathbb{R}[[x]]$, with $\phi(0)=1$, we have defined $g(x):=\int^x_0{dt\over \phi(t)}$, $f:=g^{-1}$ and $$F(x,y)=f\big(g(x)+g(y)\big)=\sum_{n=0}^\infty \psi_n(x) {y^n\over n!}\in\math …
- 60.6k
16
votes
The function $\sum_{0}^{\infty} x^n/n^n$
I wish to add a remark to Noam D. Elkies' beautiful answer. From the integral representation for $f$, putting $e^{-t}=s$ in the integral,
$$f(-x)=1-x\int_0^\infty e^{-xte^{-t}} e^{-t}dt = 1-x\int_0^1 …
- 60.6k
16
votes
Positivity of a finite sum involving Stirling numbers
The numbers $a_{n,m}$ are in fact the Fourier coefficients of the polynomial
$$P_n(x)=\prod_{j=1}^{n-1} \Big( \frac{nx}{2} + \frac{n}{2}-j\Big) $$
with respect to the Chebyshev measure $d\sigma:=(1-x^ …
- 60.6k
15
votes
Accepted
Convergent subsequence of $\sin n$
As to the convergenge to zero: note that the convergents of the continuous fraction for $\pi$ provide a rational approximation $|\pi - p_n/q_n| < 1/q_nq_{n+1}$ so that $\sin p_n\to 0$.
The sequence o …
- 60.6k
13
votes
Where in mathematics do these polynomials appear?
Not a truly satisfying answer, but maybe it puts things under a slightly more natural view. Consider the linear map $L$ on the space $k[x]$ such that $Lp(x):=p(x^2)$ . So $(L-I)^k$ expands by the bin …
- 60.6k
12
votes
Limit associated with complementary sequences
Let $\alpha_*$, $\alpha^*$ denote the lower, respectively upper asymptotic density of the set $A$, and $\beta_*$, $\beta^*$ the lower and upper asymptotic density of the set $B$. Note that $$\limsup …
- 60.6k
10
votes
Sums of arctangents
I have no references for this particular series, but here's some hints to get a closed formula for the coefficients listed above by Michael Renardy.
If we let $u:=\frac {1-x} 2$, an expansion
$$\arct …
- 60.6k
10
votes
Accepted
How to show this series converges $\sum\limits_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2)$
As indicated by Todd Trimble in comments, we can use the Dirichlet test; here, since $$\sin(n)\sin(n^2)=\frac12\big( \cos n(n-1) - \cos n(n+1) \big)$$
we have a telescopic sum $$\sum_{n=1}^M \sin(n) …
- 60.6k
9
votes
Accepted
On the finite sum of reciprocal Fibonacci sequences
We need to prove, equivalently
$$\frac1{F_{2n-1}+1}<\sum_{k=n}^{2n}\frac1{F_{2k}}\le \frac1{F_{2n-1}}, $$
that is, by the above expression for $F_{k}$, since $\beta=-\alpha^{-1}$, we need to check the …
- 60.6k
8
votes
Non-arithmetic proof of the integrality of a rational expression
The generating function $g(x):=(1−k^2 x)^{-1/k}$ satisfies, besides $g(0)=1,$
$ g^k= 1 + k^2 x\ g^k $,
whence we may express $c(k,n)$ as a sum of products of $c(k,j)$, with $j < n$, showing induc …
- 60.6k
8
votes
Uniqueness of Neumann series
edit: This was meant to recall a first elementary but relevant fact that was not mentioned at all, that is orthogonality.
Bessel functions $\{J_n\}_{n\in\mathbb N_+}$ are orthogonal on $\mathbb R$ w.r …
- 60.6k
7
votes
Radio-playing sequence
Although we have by now a precise answer, I'd like to keep the summer mood of the question and play a little more with it by an elementary arithmetic approach.
The solution I wish to sell is good for …
- 60.6k
7
votes
Accepted
In search for a counterexample related to the Abel-Stolz theorem
In fact a trivial counterexample to the question, as now clarified, is just $a_n:=(-1)^n$ with $s:=\frac12$. In your notation, $a_n\to\frac12\;\bf (A)$, because the series $\sum_{n=0}^\infty (-z)^n$ c …
- 60.6k