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 1131
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.
123
votes
Accepted
Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?
Looks like the computers really spoiled us :)
GH gave a perfectly valid answer already but the cheapest way to prove positivity is to write $\int_0^1(1-t^n)\log(\frac 1t)^{-3/2}\,\frac{dt}t=c\sqrt n …
- 61.9k
33
votes
Accepted
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
OK, here is a fairly simple proof that for any positive integer $n$ and any positive real numbers $x_1,\ldots,x_n$,
$$ \sum_{i,j=1}^{n}\left\{\frac{x_i}{x_j}\right\}\leq \frac{9}{14}n^2\,.$$
That the …
- 61.9k
17
votes
Accepted
Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\pr...
If you do not require monotonicity of $f$, the construction is pretty simple and is a combination of a few facts we normally (should) teach in elementary number theory and Fourier analysis classes. Ho …
- 61.9k
16
votes
Accepted
Convergency radius of the generating series for A93637
This is pretty simple, really. Note that we can obtain our power series in the following way. Define on (formal) power series with positive coefficients the transform
$$
T\sum_{k=0}^\infty b_kx^k=\tex …
- 61.9k
11
votes
Accepted
Given four conditionally convergent series, is there a single sequence of naturals such that...
There is a counterexample with 4 series. Notice that the problem is equivalent to asking if for every sequence of vectors $X_j$ in $\mathbb R^4$ with lengths tending to $0$ and the infinite sum of abs …
- 61.9k
10
votes
Accepted
If two functions are equal to their Newton series, is their composition also equal to its Ne...
Here goes, as promised.
Let $f$ be entire of order less than $1$, so $|f(z)|\le Ce^{|z|^p}$, $p<1$. Write the Newton polynomial
$$
P(x)=\sum_{k=0}^n\Delta^kf(0) {x \choose k}
$$
Note that $g(k)=f(k …
- 61.9k
10
votes
Accepted
The digit sum: $s(na)=s(nb)$
OK, by Seva's request I'm getting somewhat more serious :) Fix $a$, $b$. Take large $M$ to be chosen shortly. Take a $3MN$ digit number $n$ (something from $0$ to $10^{3MN}-1$, written with head zeroe …
- 61.9k
8
votes
An inequality involving $k$-generalized Fibonacci numbers
$F_{\ell+1}$ is just the number of ways to tile an interval of length $\ell$ by tiles of lengths up to $k$. Now take the interval of length $2m-3$ and look at what happens at the mark $m-2$. You may h …
- 61.9k
8
votes
Accepted
On a monotonicity property of Fourier coefficients of truncated power functions
I prefer to write $2\pi$ in the exponent and reverse the direction of the monotonicity (none of which really matters, of course), so we shall fix $k>1$ (real in general), $n\in\mathbb Z_+$ (that is es …
- 61.9k
7
votes
Accepted
Coefficient bounds of an inequality
This is true.
I prefer to denote $q_i=p_i^{-1}$, $\alpha_i=a_ip_i$, $\beta_i=b_ip_i$, $A_i=\sum_{j=1}^i\alpha_j$, $B_i=\sum_{j=1}^i\beta_j$. Now we have to check that
$$
\sum_i q_i\beta_iA_i^k\le C\ …
- 61.9k
7
votes
On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1...
I am not quite sure if the previous long discussion has already resulted in a full proof of anything but here is the crude bound that shows that the expectation in question is infinite.
Let $X_i$ be i …
- 61.9k
6
votes
Two Equal Series?
Here is a link to a counterexample for conditionally convergent complex series.
- 61.9k
6
votes
Accepted
Pseudo-alternate series
That would be more than welcome on AoPS, College Playground. For MO, it is hardly appropriate.
The statement is always true. Start with the fact that if the sums $\sum_{i=k}^m b_i$ ($1\le k\le m\le …
- 61.9k
6
votes
Accepted
Ratio of Sequences Sum Inequality
Here is an old cheap trick that may be helpful. Assume that $\sum_i a_i=A$, $\sum_i b_i=B$ and $a_i\le b_if(b_i)$ where $f$ is a decreasing function tending to $0$. Choose $b$ so that $f(b)=\frac{A}{2 …
- 61.9k
6
votes
Accepted
Inequality for functions on [0,1], continued
OK, here goes.
We start with changing the notation ($z\to 20z^2$, $-z-3\to r$, $20rz\to y$ means that what was denoted by $z$ will be denoted by $20z^2$ from now on, $r$ is $-z-3$ with new $z$, so it …
- 61.9k