2
votes
1answer
56 views

Eisenstein-type series

Is the series, $$1 - 24\sum_{n = 1}^\infty \frac{q^{2n}}{(1 - q^{2n})^2},$$ somehow related to $$E_2(q) = 1 - 24\sum_{n = 1}^\infty \frac{nq^n}{1 - q^{n}},$$ the Eisenstein series of weight 2?
3
votes
0answers
34 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
6
votes
1answer
143 views

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
0
votes
1answer
14 views

Some questions about series and generalized integrals

Do someone of you know any good site about generalized integrals and series (convergence and divergence) with example exercises with detailed solutions ? I would be happy if you did know any. My ...
1
vote
0answers
41 views

Good source of worked examples for Testing Infinite Series

I'm looking for a source/s with a large number of worked examples in Testing Infinite Series using the following methods - Divergence Test Comparison Test and P Test Ratio Test Alternating ...
5
votes
1answer
143 views

On the possible values of $\sum\varepsilon_na_n$, where $\varepsilon_n=\pm1$ (i.e., changing signs of the original series)

I have used the following result in an answer on this site. Suppose that $a_n>0$ are positive real numbers such that $\sum\limits_{n=1}^\infty a_n=+\infty$ and $\lim\limits_{n\to\infty} ...
2
votes
0answers
44 views

References on the Evaluation of Series

What are the best introductory resources for learning how to evaluate tricky series? I don't have any specific series in mind, I am looking for general methods to solve a variety of series. The ...
3
votes
1answer
38 views

If $J:{\Bbb N}\to{\Bbb N}\times{\Bbb N}$ is a bijection, then $\sum_{n=1}^\infty a_{J(n)}=\sum_{i=1}^\infty\sum_{j=1}^\infty a_{ij}$

...assuming that $a_{ij}$ are nonnegative reals and the sum is convergent. Can someone give me a proof or reference for this theorem? I remember reading it in my textbook once, but now I forget and it ...
3
votes
1answer
41 views

two natural integer sequences

Have the integer sequences $f$ and $g$ defined below already been studied? Do they have any interesting property? Define the maps $f,g:\mathbb Z_{ > 0}\to\mathbb Z_{ > 0}$ as follows: Define ...
1
vote
2answers
64 views

Sequences that look like they have integer terms

I've read somewhere about a sequence that is an integer for many terms but eventually loses the integer property. I remember it to be an April fool's joke in a problem column of some kind. I'd like to ...
2
votes
1answer
53 views

Any insight about this sequence of numbers?

I don't have a background in math beyond high-school calculus and one course in discrete math. I was hoping you guys might be able to give me some information about the sequence of numbers generated ...
1
vote
0answers
58 views

$\prod_{n}f_{n}$ converges uniformly $\Rightarrow $ $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly

Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$. Then is it true that ...
1
vote
0answers
90 views

Summary of divergent series summation methods and relations between them?

There are a number of methods of assigning sums to series that do not necessarily converge, e.g. Cesàro summation, Abel summation, Ramanujan summation, etc. (There is also the trivial method of only ...
15
votes
1answer
309 views

What is known about doubly exponential series?

I've been exploring functions that have a general form: $$\sum_{k=0}^\infty{ a^{b^k} } \tag{1}$$ In particular, I'm now checking this equality, which seems to hold: $$2 \sum_{k=0}^\infty{ \left( ...
0
votes
0answers
65 views

What was Euler's misconception about functions and infinite series?

I just read this on Strichartz' The way of Analysis: [...] Euler - the leading mathematician of the eighteenth - developed all techniques needed for the study of Fourier series, but he never ...
2
votes
2answers
117 views

Where can I find out more on Collatz-conjecture like sequences?

I'm interested in Collatz-conjecture (the 3n+1 problem) like sequences. I'm interested in any literature that contains information about problems that are divided into similar cases. I'm ...
3
votes
2answers
50 views

Relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ [duplicate]

What is the relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ where $a_{n} \in \mathbb{C} \ \forall n$ ? Where can I find some references about this topic ?
1
vote
1answer
136 views

Does the Gauss' Trick Really Belong to Gauss?

We all have heard the story of the young Guass, summing 1 to 100 by writing the sum backward below the original one. In this article, just two books are referred for the trick. I looked at both of ...
1
vote
1answer
49 views

discrete mathematics , sequences, characteristic equation

Today in my discrete mathematics class we started combinatorics and also solved some some recurrence relation sequences using the characteristic equation. So my question is can you guys point me to ...
5
votes
1answer
254 views

Series involving Logs

I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g. $$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$ I realise that this comes from $x^y ...
3
votes
1answer
463 views

Cesaro summable implies Abel summable

I've looked through the Stack old questions, and searched the net, and I haven't found a proof that Cesaro summability implies Abel summability. Is the proof extremely complicated? Does anyone know a ...
6
votes
1answer
79 views

percentage of numbers starting with $2$ in $\{2^n\}$

I have once heard a professor telling (during a course on Fourier theory) that there is a way to determine the numbers starting with a $2$ in the sequence $\{2^n\colon n\in\mathbb{N}\}$. I asked him ...
2
votes
2answers
98 views

sources for all infinite series?

I'm looking for a source for all of the known infinite series, if such a source exists. The following site is the best source that I have found so far, but I do not know if it is complete and it does ...
4
votes
1answer
92 views

What is the name for defining a new function by taking each k'th term of a power series?

With the definitions of the three functions $$ f(x)= 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + ... \\ g(x)= x + \frac{x^4}{4!} + \frac{x^7}{7!} + ... \\ h(x)= \frac{x^2}{2!} + \frac{x^5}{5!} + ...
1
vote
0answers
77 views

Literature leading up to algebraic topology

First and foremost, allow me to apologise for the quite general and perhaps vague nature of this question and for the presence of any misunderstandings in the statements I will use. I know very well ...
0
votes
1answer
62 views

What is the name of this theorem?

I haven't had a formal introduction to series and summation, but I have read somewhere that to find out if a sum converges/diverges, you can take the integral and if the integral has a limit when ...
0
votes
0answers
122 views

fractional moments of binomial distribution

I would appreciate your help in learning about the quantity $$ \large\sum_{j=0}^J {J \choose j} p^j (1-p)^{J-j} j^\alpha, $$ for any $\alpha > 0$, but in particular for $\alpha\in (0,1)$. Which ...
2
votes
0answers
77 views

Name for this Sum

This is probably a very basic question, but I am asking because I could not find the answer online. I have been trying to find out some properties regarding the following sum ...
0
votes
2answers
95 views

Changing the index of a summation - what is it called?

About $\sum$ (summation), what is it called when you change the index of a summation? My teacher does it all the time and I just don't get it! Please send some links so I can learn it.
5
votes
1answer
110 views

Introductory book on series

I'm looking for a good introductory book on sequences and series (mathematics). Do you have any advice?
2
votes
0answers
67 views

To find the limit of three terms mean iteration

We know that the arithmetic-geometric mean $AGM(a,b)$ of $a$ and $b$ defined as $$2a_1=a+b$$ $$b^2_1=ab$$ $$2a_n=a_{n-1}+b_{n-1}$$ $$b^2_n=a_{n-1}b_{n-1}$$ $AGM(a,b)=\lim\limits_{n\to \infty} ...
-2
votes
1answer
111 views

Why multiple sequences and series are ignored?

By a multiple sequence of order $n\in\mathbb{N}^*$ I mean a function $a:~\mathbb{N}^n\longrightarrow\mathbb{K}~(=\mathbb{R}\text{ or }\mathbb{C})$ such that $$a(i_1,\cdots,i_n)=a_{i_1i_2\cdots a_n}$$ ...
16
votes
1answer
480 views

How can we prove $\pi =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\cdots\,$?

I saw the beautiful result that was proved by Euler in Wikipedia but I do not know how it can be proved. $$\pi =1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6} + ...
4
votes
0answers
67 views

What's the name for this unimodal sequence?

Let $a_0, a_1, \ldots, a_n$ be an increasing sequence of positive numbers, and consider the sequence $s_1,\ldots,s_n$, where $$ s_k \;=\; \frac{a_0+\cdots + a_k}{k}. $$ So $$ s_1 \;=\; a_0+a_1,\quad ...
1
vote
1answer
136 views

A sequence of integral tends to zero

Suppose that $a<b$ are two fixed real numbers, and $g_n$ is a sequence of real functions on $[a,b]$. What about good conditions approximately equivalent to the following proposition: ...
2
votes
3answers
96 views

What type of Hypergeometric series is this?

I am trying to find a closed form for the series $$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$ $m$ is a nonzero positive integer, and $b$, ...
71
votes
4answers
2k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
6
votes
1answer
125 views

Umbral calculus with negative indices (and powers)

Can we do umbral calculus with negative indices (and powers)? Can we write $a_{-n} \equiv a^{-n}$ or $L[a_{-n}] = a^{-n}$ where $L$ is a linear functional and $n$ need not be negative? The common ...
1
vote
1answer
30 views

Searching for a certain sequence

How do I search OEIS for a certain sequence? For example I want the number of ways $1, 2, 3,\ldots, n$ can be arranged so that the numbers in the even places are greater than their neighbours.
4
votes
1answer
117 views

series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

I found an interesting series $$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( ...
13
votes
3answers
356 views

Has someone seen a discussion of the (divergent) summation of $\sum\limits_{k=0}^\infty (-1)^k (k!)^2 $?

In extending my studies of the Eulerian matrix and its suitability for a matrix-based divergent summation procedure I'm trying to proceed to sums of the form $$ S = \sum_{k=0}^\infty (-1)^k (k!)^2 $$ ...
3
votes
2answers
128 views

Sequence and series problems with slick group theoretic solutions.

So I'm trying to get this unmotivated student in my class interested in learning group theory. I've recently ascertained that he likes analysis a bit better than algebra, and that sequences and ...
16
votes
2answers
343 views

How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?

I'm considering the transfer-function $$ t(x) = \log(1 + \exp(x)) $$ and find the beginning of the power series (simply using Pari/GP) as $$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
5
votes
1answer
360 views

A theorm about Cesàro mean, related to Stolz-Cesàro theorem

Original Title: Tauberian theorems and Cesàro sum Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14) $\newcommand\abs[1]{\left\lvert#1\right\rvert}$ If $\{s_n\}$ ...
2
votes
0answers
123 views

Double sequences and series double.

I would like a good reference devoted exclusively to double sequences and series double. At the moment I have only this reference in google. Thanks in advance.
4
votes
1answer
78 views

A reference for some fact in analysis

I am looking for a reference for the following fact. Any hints would be appreciated. Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ ...
0
votes
1answer
68 views

Convergence of series of elementary symmetric functions

Let $x_1,x_2,x_3,\ldots$ be an infinite sequence of real numbers (or assume they're complex numbers if you find that convenient). Let $e_0,e_1,e_2,e_3,\ldots$ be the elementary symmetric functions of ...
2
votes
0answers
58 views

tangents of infinite sums — reference request

This section of Wikipedia's List of trigonometric identities was, as far as I know, written entirely by me. For a time, it dealt only with finite sums, and said something like this: $$ ...
1
vote
2answers
79 views

About the sequence $\{\sin (n^2) \}_n$

A colleague of mine asked me if the behavior of the sequence $\{\sin (n^2) \}_n$ is known. In particular, does it converge? If not, what are its liminf and limsup? I had to admit that I cannot answer ...
1
vote
1answer
55 views

How to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?

Is there some standard way to approximate a complex linear homogenous recurrence with constant coefficients with a simple one? For example, I might want to approximate $$ ...