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Results tagged with sequences-and-series
Search options answers only
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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.
6
votes
Proving convergence of solution of a fixed point equation
Convergence to $0$ is trivial: as you noticed yourself, the coefficient is just ${n-1\choose k,k,n-1-2k}$. Splitting the rest as $(\sqrt\alpha x)^k(\sqrt\alpha x)^k(1-x)^{n-1-2k}$, we can bound $f_n(x …
- 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
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
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
3
votes
Why is the following recurrent sequence convergent?
That is, indeed, more bark than bite, but the bark is somewhat louder than mlk presented it.
First of all, let us define $X_k=\max(|x_k|,|x_{k-1}|)$ and notice that ${2n+1\choose 2k+1}\ge 2^{\min(k,n- …
- 61.9k
6
votes
Accepted
Unconditionally convergent series in $\ell_2$ consisting of $\ell_1$-small vectors
If I understand correctly what you are asking, the answer is "certainly not".
Consider $k$ orthogonal vectors $v_j$ with $k$ coordinates $\pm 1$ (the Hadamard matrix). Now multiply them by $t$ and tak …
- 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
5
votes
Accepted
Convergence for a non-linear second order difference equation
I would suspect that non-classical arguments are needed to do so.
All you need to know is that $t\mapsto \frac 1{1+t}$ is a decreasing function, so for $0<x\le x'$ we have $\frac{\log(1+x')}{\log(1+x) …
- 61.9k
5
votes
Accepted
The sum of a series
I have no doubt that you have figured this exercise out by now but let me post the solution just for the fun of it.
Denote $U_k=\prod_{m=1}^k\frac 1{q^m-1}$ with the usual convention $U_0=1$.
We hav …
- 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
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
3
votes
Approximate sum by an integral: valid or not?
The main observation:
Let $f,g$ be analytic in the disk $\{|z|\le 2\delta\}$ and real on the interval
$(-2\delta,2\delta)$. Assume that $f(0)=0$ and $f(x)<0$ for $0<|x|<2\delta$. Let $\psi$ be any $C …
- 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