For questions about recurrence relations, convergence tests, and identifying sequences

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7
votes
1answer
231 views

Closed form for the series $\sum\limits_{k=0}^\infty \frac{(-1)^k\exp(-\lambda(2k+1)^2)}{(2k+1)^3}$

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
220 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
152 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
576 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
204 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 ...
2
votes
3answers
3k 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
39 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: ...
17
votes
3answers
698 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
68 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
322 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?
1
vote
2answers
544 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
460 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
73 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
136 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
83 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
73 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
448 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
170 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
110 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
161 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
344 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 ...
6
votes
5answers
3k 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
540 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
100 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 ...
5
votes
2answers
237 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
2k views

Finding the sum of this alternating series with factorial denominator.

What is the sum of this series? $$ 1 - \frac{2}{1!} + \frac{3}{2!} - \frac{4}{3!} + \frac{5}{4!} - \frac{6}{5!} + \dots $$
1
vote
2answers
199 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
289 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
158 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
63 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
53 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
71 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
363 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
47 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 ...
5
votes
0answers
223 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
967 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
74 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 ...
4
votes
4answers
8k 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
1k 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 ...
8
votes
2answers
239 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 ...
1
vote
1answer
248 views

Using the sum of squares formula to solve more complex sums.

I'm studying integration and trying to figure out how to use the sum of squares formula to solve more complicated sums. For example: knowing that $$\sum_{i=1}^n i^2 = \frac{n (n+1) (2 n+1)}{6}$$ ...
5
votes
2answers
864 views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
0
votes
2answers
75 views

Is there a good description of the series $\sum_{j=0}^{n}a^j b^{n-j}$?

To what does the serie $\sum_{j=0}^{n}a^j b^{n-j}$ converge? Does this serie have a name?
0
votes
1answer
95 views

Convergence of an ugly series

I am wondering for $n > 1$ if the following series converges and if possible what it equates to. $$\sum _{m=0}^{\infty }\dfrac {\log\left( \dfrac {\left( n+m-1\right) !} {\left( n-1\right) ...
20
votes
4answers
905 views

Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?

I would like to evaluate the sum $$ \sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right) $$ Where $\operatorname{Si}$ is the sine integral, defined as: $$\operatorname{Si}(x) := ...
2
votes
2answers
143 views

A modification of the harmonic series that causes it to converge. [duplicate]

Possible Duplicate: sum of inverse of numbers with certain terms omitted We are working in base $b$, a positive integer greater than $1$. Let $S$ be the set of all positive integers that ...
15
votes
3answers
845 views

Convergence\Divergence of $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$

Prove convergence\divergence of the series: $$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$ Here is what I have at the moment: Method I My first way uses a result that ...
2
votes
4answers
340 views

can one derive the $n^{th}$ term for the series, $u_{n+1}=2u_{n}+1$,$u_{0}=0$, $n$ is a non-negative integer

derive the $n^{th}$ term for the series $0,1,3,7,15,31,63,127,255,\ldots$ observation gives, $t_{n}=2^n-1$, where $n$ is a non-negative integer $t_{0}=0$
1
vote
4answers
5k views

Does $\sum\ln(n) / n^3$ converge?

By Direct Comparison Test: $$\ln(n)/n^3 < n/n^3 = 1/n^2 $$ Converges as that is a convergent a p-series By n'th term Test: $$(\ln(n))' = 1/n$$ $$(n^3)' = 3n^2$$ $$\lim_{n\to\infty} (1/n) / ...
11
votes
3answers
357 views

Collatz-ish Olympiad Problem

The following is an Olympiad Competition question, so I expect it to have a pretty solution: For a positive integer $d$, define the sequence: \begin{align} a_0 &= 1\\ a_n &= ...