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Results tagged with sequences-and-series
Search options answers only
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user 297
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.
31
votes
Elementary proof of the equidistribution theorem
There is a really easy proof that just uses the Pigeonhole principle. Let $\alpha$ be irrational.
Lemma: For any $\delta >0$, there is an $n>0$ such that $(n \alpha) \in (- \delta, \delta) \setminus \ …
- 156k
26
votes
Accepted
"Are you taller than the average of those who are taller than the average?"
As in Nate's answer, we are interested in iterating the function
$$G(y) := \frac{ \int_{y}^{\infty} x e^{- x^2} dx}{\int_{y}^{\infty} e^{- x^2} }.$$
The numerator is $e^{-y^2}/2$ (elementary). The de …
- 156k
24
votes
Rearrangements that never change the value of a sum
For the purpose of recording an answer rather than just a pile of links: Michael Hardy requires that
if either limit exists then so does the other and in that case then they are equal?
Let's cal …
20
votes
Does this sequence always give an integer?
This is the special case $(p,q,r)=(1,2,3)$ of the $3$-term Gale-Robinson recurrence:
$$x_{n+p+q+r} x_n = x_{n+p} x_{n+q+r} + x_{n+q} x_{n+p+r} + x_{n+p+q} x_{n+r}$$
Fomin and Zelevinsky proved that, …
- 156k
18
votes
Freeness of a Z[x]-module
It is not free. Set $f(x) = x(x-1)(x-2)(x-3)/2$.
Claim: $f(x)$ is in $R$.
Proof: We have
$$\frac{f(x+N)-f(x)}{N} = \frac{N^3+11 N}{2} + (\mbox{an element of } \mathbb{Z}[x,N]).$$
The fraction $(N^3 …
- 156k
10
votes
The coefficient of a specific monomial of the following polynomial
The coefficient is always zero. I use the superb conventions of Concrete Mathematics: Sums are over all integers unless otherwise indicated, and $\binom{n}{k}$ is $0$ if $k<0$ or $>n \geq 0$.
Expandi …
- 156k
10
votes
A possible surprise involving Euler's constant $e$
Sure.
$$c_n = \frac{1}{n+1} + \frac{1}{(n+1)(n+2)}+ \frac{1}{(n+1)(n+2)(n+3)} \cdots$$
so
$$\frac{1}{n+1} < c_n < \frac{1}{n+1}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots = \frac{1}{n}.$$
That prov …
- 156k
9
votes
Optimal Talmudic Zigzag
This is a special case of finding the longest path in a directed acyclic graph. Namely, the vertices of our graph are $1$, $2$, ..., $n$, there is an edge $i \to j$ for each $i<j$ and the length of th …
- 156k
9
votes
Reciprocals of Fibonacci numbers
Just to make the obvious comments:
Let $q = (1 - \sqrt{5})/2$. Then $F_n=((-q^{-1})^{n} - q^n)/\sqrt{5}$. So
$$\sum \frac{1}{F_n} = \sqrt{5} \sum \frac{q^n}{1-(-1)^n q^{2n}}.$$
This looks kind of like …
8
votes
When is there a closed form for $\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$?
As GH from MO says, it depends what you think is a closed form.
Any function of the form $P(n)/Q(n)$ with $\deg P \leq \deg Q -2$ can be written as a linear combination of $1/(n+1) - 1/(n+\alpha)$ an …
- 156k
8
votes
Accepted
Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' differen...
UPDATE I now have a complete proof. I'll start with the Pascal's triangle case.
For any sequence of real numbers $r=(r_1, r_2, \ldots, r_k)$, define the number of sign changes of $r$ as follows: Dele …
- 156k
8
votes
Slick proof of Stirling's Formula?
I've played around with this a bit. I have a slick lower bound, but not a slick upper bound.
We start with the $ \Gamma $-integral:
$$
n! = \int_{x=0}^\infty x^n e^{-x} dx = \int_{y=-n}^\infty (n+y)^n …
- 156k
7
votes
Accepted
Angle between two given vector is small. Can we permute coordinates of them such that new ve...
No. Let
$$\vec{x} = \vec{y} = \frac{1}{\sqrt{12+6 \sqrt{3}}} (-2-\sqrt{3},1,1+\sqrt{3}).$$
So $\vec{x} \cdot \vec{y} = 1$, but $x_1 y_2+x_2 y_1 + x_3 y_3=0$.
- 156k
6
votes
Accepted
What is the rate of convergence?
Putting $x_n = 1+y_n$, we have $y_{n+1} = y_n + \frac{y_n^2}{2}$. There is a standard method for analyzing recursions of the form $y_{n+1} = y_n+O(y_n^2)$. A good introduction is chapter 8.4 in de Bru …
- 156k
6
votes
If two functions are equal to their Newton series, is their composition also equal to its Ne...
Another counter-example is extractable from Gerald Edgar's answer to this question, where he shows that $\sin (ax)$ is discrete analytic for $a \in (-\pi/3, \pi/3)$. So take $
f(x) = \sin ((\pi/4) x)$ …
- 156k