Recurrence relations, convergence tests, identifying sequences
2
votes
0answers
46 views
Is exponent of discrete-analytic function also discrete-analytic?
Lets define a discrete analytic function such a function that is equal to its Newton series:
$$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$
Is function $g(x)=e^{f(x)}$ also ...
5
votes
1answer
143 views
$n$-th digit in the sequence of natural numbers
What's the $n$-th digit in the sequence $S$ of numbers formed by the natural numbers, i.e.,
$n$-th digit in the sequence 1 2 3 4 5 6 7 8 9 10 11 12... ? For example 11th digit in the sequence is 0.
-2
votes
2answers
124 views
Real number in terms of infinite series
This is just continuation of my previous post.
$$ A = a_0 + \left(\frac{1}{a_1}\right)^k + \left(\frac{1}{a_2}\right)^k + \left(\frac{1}{a_3}\right)^k +\ldots$$
Where $i\ge1$ and the recurrence ...
2
votes
1answer
38 views
How to compute the series which seems different with the questions I raised before?
How to compute the series which seems different with the questions I raised before?
$\lim_{N\rightarrow\infty}N\sum^{N}_{k=2}\left(\frac{k-1}{N}\right)^{N^2}$
0
votes
2answers
69 views
Show that the integral of this rational function is equal to an infinite alternating harmonic series
One of my friends gave me the following question from his review, I have little experience to dealing with these types of questions in Analysis so if you could help us just to get started it would be ...
0
votes
0answers
115 views
How to prove that $(1^3+2^3+\cdots+n^3)=(1+2+\cdots+n)^2$ [duplicate]
Possible Duplicate:
Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$
How can one prove that ...
8
votes
2answers
196 views
Techniques for summing ratio of binomial coefficients
There are several identities that involve the sum of the product of binomial coefficients. However what I am searching for is an identity that involves the ratio of binomial coefficients. ...
1
vote
1answer
214 views
The coefficients in the inverse of unit lower triangular Toeplitz matrix
I have a question about the inverse of lower triangular Toeplitz matrix.
There is a matrix $Q$ which can be infinite-dimensional.
$$
Q=\left[\begin{array}{ccccc}
1\\
-k_{1} & 1\\
-k_{2} & ...
2
votes
0answers
99 views
Convergence of a function in a metric space to its metric
Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
1
vote
3answers
103 views
Closed form for $T(1) = K, T(x) = xT(x-1) + x$?
I'm looking for a closed form for the following recurrence:
$T(1) = K$
$T(x) = xT(x-1) + x$
I know it is factorial-like but I am unable to get an exact answer.
6
votes
1answer
157 views
sum of infinite series
Does there exist an explicit expression for
$$\sum_{k=0}^{\infty }\frac{\left( -1\right) ^{k}e^{-\lambda \left(
2k+1\right) ^{2}}}{\left( 2k+1\right) ^{3}}\;,$$
where $\lambda$ is a positive scalar? ...
6
votes
1answer
157 views
To show Taylor series of a Fourier transform $\hat{f }$ converges to $\hat{f}$
I got some trouble with the following question.
Say $f$ is in $L^1(R)$ with compact support . I need to show (1) $\hat{f(\zeta)}$ is infinitely differentiable and all derivatives are continuous. ...
-1
votes
1answer
133 views
Real Numbers expressing in terms of series
From the literature, I have found the following:
Any real number A (say) can be expressed as
$ A = a_1 + (1/a_1) + (1/a_2) + (1/a_3) +\ldots$
Where $a_1\ge2$ and the recurrence relation $a_{i+1}\ge ...
3
votes
1answer
236 views
Subsequential limit of sequence
I'm trying to determine all subsequential limit points of the following sequence
X_n = cos(n)
Not sure how to decompose this into subsequences.
Anyone know how ...
3
votes
3answers
143 views
Does the series $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverge
Is there a handy way to tell if $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverges or not? I have a hunch that it diverges, since it looks like the sum is just $\zeta(1)-1=\infty$. But ...
1
vote
3answers
710 views
limit of a sequence $(-1)^n$ instead $1$
I should calculate the limit of a sequence. A friend told me, that the solution is $1$. But I got $(-1)^n$.
The exercise is: $\lim\limits_{n \to \infty} \frac{1}{n^2} + (-1)^n \cdot ...
1
vote
1answer
36 views
What is $\phi(k)$ in $\sum_{k=1..n} \phi(k)\lfloor n/k \rfloor^2$?
I would like to compute a general $n$-term of a sequence
$$ 1, 5, 12, 24, 37, 61, 80, \dots$$
However I do not understand what $\phi$ refers to in the formula at http://oeis.org/A018806:
...
16
votes
3answers
598 views
A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)
Is there a known geometric proof for this famous problem? $$\zeta(2)=\sum_{n=1}^\infty n^{-2}=\frac16\pi^2$$
Moreover we can consider possibilities of geometric proofs of the following identity for ...
2
votes
1answer
66 views
How to compute this limit related to series?
How to compute $\lim_{N\rightarrow+\infty}\frac{\ln^2N}{N}\sum_{k=1}^{N-1}\left(\frac{k}{N}\right)^{2\ln N-2}$? thank you.
0
votes
2answers
208 views
machine learning project ideas
I am interested about playing with machine learning algorithm and time series analysis. Is there website/resource with a comprehensive list of sample projects/proposals one may be interested about?
0
votes
2answers
197 views
Use of ratio test and comparison test to determine converging series
Which of the following series converge, use ratio or comparison test to demonstrate:
$\sum_{n=1}^{\infty} \dfrac{n}{n^2+\cos^2(n)}$
$\sum_{n=1}^{\infty} \dfrac{(-10)^n}{4^{2n+1}(n+1)}$
...
1
vote
1answer
220 views
Convergence in $\sup$ norm $\Rightarrow$ Cauchy in $\sup$
Let $(f_n)$ be a sequence of bounded functions on a set $E \subseteq \mathbb R$ and suppose that $f$ is a bounded function such that $\|f_n - f\|_{\infty} \to 0$ as $n \to \infty$. Prove that $(f_n)$ ...
0
votes
1answer
58 views
Limit of expression involving exponentials.
For $0 \le s < 1$, $t \ge 0$ let
$$G(s,t) := \frac{e^{-t} s}{\sqrt{1-(1-e^{-2t})s}}$$
For $\lambda > 0$ compute the limit of $G(e^{-2\lambda e^{-2t}},t)$ as $t \rightarrow \infty$.
7
votes
2answers
126 views
How to solve this limit related to series?
How to solve the following limit?
$$\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{k=1}^{N-1}\left(\frac{k}{N}\right)^N$$
1
vote
1answer
71 views
Proof for length of period in simple modulo $N$ sequence.
I am looking for a concise proof that the length of the smallest period of the sequence
$$f[n] = a n \pmod N $$
is $N$ if $(a,N) = 1$. From the Pigeonhole Principle, it is not hard to show that ...
3
votes
1answer
63 views
Change domain on series - Counting aplitudes
If I have a function
$$f(t) = y$$
where $t$ & $y$ are positive Integers
for $t = \{1,2,3,4,5,6,7,8\} \to y = \{1,1,1,2,1,2,3,1\}$
How can I create a function $g(y)$ such that it counts the ...
1
vote
2answers
196 views
How to approximate the value of $ \arctan(x)$ for $x> 1$ using Maclaurin's series?
The expansion of $f(x) = \arctan(x)$ at $x=0$ seems to have interval of convergence $[-1, 1]$
$$\arctan(x) = ...
3
votes
4answers
158 views
How to show that $\sum\limits_{n=1}^\infty \exp(i\,nz)$ converges?
How do I show that $$ f(z) = \exp(i\, z) + \exp(i\, 2z) + \ldots + \exp(i\, nz) + \ldots $$ converges? Problem is taken from a Yahoo! Answers question: "Find the infinite sum of sin(n)/n?".
0
votes
1answer
60 views
Relationship between different sequences generated through modulo arithmetic.
I am unsure the formal mathematical terminology/notation for dealing with sequences generated from integer modulo arithmetic. So first off, could someone recommend a book that focuses on the ...
5
votes
1answer
112 views
Radius of Convergence of this Series
This is a question from a GRE math subject test practice material.
$$ \sum^{\infty}_{n=1} \frac{n!x^{2n}}{n^n(1+x^{2n})} $$
The set of real numbers $x$ for which the series converges is: $\{0\}$, ...
4
votes
3answers
177 views
Writing 1/3 as a sum of other numbers
Is it possible to write $0.3333(3)=\frac{1}{3}$ as sum of $\frac{1}{4} + \cdots + \cdots\frac{1}{512} + \cdots$ so that denominator is a power of $2$ and always different?
As far as I can prove, it ...
3
votes
3answers
1k views
Sum of series $\sin x + \sin 2x + \sin 3x + \cdots $
Please help me compute the sum of the series:
$$\sin(x)+\sin(2x)+\sin(3x)+\cdots$$
1
vote
2answers
122 views
Fast variance calculation
Suppose to have a sequence $X$ of $m$ samples and for each $i^{th}$ sample you want to calculate a local mean $\mu_{X}(i)$ and a local variance $\sigma^2_{X}(i)$ estimation over $n \ll m$ samples of ...
2
votes
1answer
81 views
Sequence of $C^1[0,1]$ functions $(f_n) \to f$ but $f \notin C^1[0,1]$
Question:
Give an example of a sequence of continuously differentiable function $(f_n)$ on $[0,1]$ so that $f_n \to f$ uniformly, but $f$ is not differentiable at all points of $[0,1]$.
My ...
4
votes
2answers
158 views
Convergence of a sequence of nonnegative real numbers $x_n$ given that $x_{n+1} \leq x_n + 1/n^2$.
Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows ...
4
votes
3answers
465 views
Finding the sum of this alternating series with factorial denominator.
What is the sum of this series?
$$ 1 - \frac{2}{1!} + \frac{3}{2!} - \frac{4}{3!} + \frac{5}{4!} - \frac{6}{5!} + \dots $$
0
votes
2answers
144 views
A similarity measure for binary sequences from a partition
I'm onto a problem about binary sequence similarity for which I have not found any existing solution. I want to share it and the approaches I have taken, although none of them convince me.
Consider a ...
2
votes
3answers
112 views
Rearrangement of sequences with limit $0$
Is it true that every real sequence that converges to zero has the property that every rearrangement of it also converges to zero?
I have a proof in mind, but I'm not 100% sure it's correct (although ...
2
votes
4answers
126 views
What is the order type of monotone functions, $f:\mathbb{N}\rightarrow\mathbb{N}$ modulo asymptotic equivalence? What about computable functions?
I was reading the blog "who can name the bigger number" ( http://www.scottaaronson.com/writings/bignumbers.html ), and it made me curious. Let $f,g:\mathbb{N}\rightarrow\mathbb{N}$ be two monotonicly ...
2
votes
1answer
49 views
Two complex series of functions
The serie of function
$$\sum_{n\in\mathbb Z}\frac{1}{{(z-n)}^2}$$
converges normally in $\mathbb C\setminus\mathbb Z$ and it defines a meromorphic simply periodic function. Now let be $\Lambda$ a ...
0
votes
1answer
48 views
Asymptotic behavior of a sequence based on a subsequence II
Let $\{a_{n}\}$ be a non-increasing sequence of positive numbers. if for some positive integers $l,p$ and $R>1$ we have $a_{(ln)^{p}}=O(R^{-n})$ as $n\to\infty$, what can we say about the behavior ...
0
votes
1answer
42 views
Rearragement of a series in Hilbert space
Let $H$ be a Hilbert space and $\sum_k x_k$ a convergent infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation ...
3
votes
3answers
271 views
Predicting the next vector given a known sequence
I have a sequence of unit vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_k,\ldots$ with the following property: $\lim_{i\rightarrow\infty}\vec{v}_{i} = \vec{\alpha}$, i.e. the sequence converges to a ...
1
vote
1answer
41 views
Can this type of series retain the same value?
Let $H$ be a Hilbert space and $\sum_k x_k$ a countable infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation ...
2
votes
0answers
108 views
Partial summation of a harmonic prime square series (Prime zeta functions)
I am trying to find the following series:
$S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$
...
2
votes
3answers
590 views
Proving that a bounded, continuous function has a supremum
Theorem: Let $f$ be a continuous real-valued function on a closed rectangle $R$ in $\Bbb R^2$. Then,
(a) $\quad f$ is bounded on $R$
(b) $\quad $There exist points $c$ and $d$ in $R$ so ...
2
votes
1answer
56 views
Sequence that maps to arbitrary positive real number
How do you construct a sequence of functions $f_n(x)$ such that
$$s = \limsup_{n\rightarrow \infty} \sqrt[n]{f_n(x)}$$
for all $s > 0$?
I know it's possible to this with a different sequence
...
1
vote
4answers
1k views
relation between integral and summation
What is the relation between a summation and an integral ? This question is actually based on a previous question of mine here where I got two answers (one is based on summation notation) and the ...
1
vote
2answers
206 views
What is linearity of Expectations?
In reading about the average case analysis of randomized quick sort I came across linearity of expectations of indicator random variable I know indicator random variable and expectation. What does ...
7
votes
2answers
223 views
How to evaluate $ \int_0^\infty {1 \over x^x}dx$ in terms of summation of series?
Is there a way to represent this integral in terms of summation of series?
$$ \int_0^\infty {1 \over x^x}dx$$
Like for example:
$$ \int_0^1 {1 \over x^x}dx = \sum_{n=1}^\infty {1 \over n^n}$$
I am not ...
