# Tagged Questions

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?
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. ...
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 ...
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 ...
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 ...
143 views

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 $$a(i_1,\cdots,i_n)=a_{i_1i_2\cdots a_n}$$ ...
480 views

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: ...
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 $b$, ...
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: ...
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 ...
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.
117 views

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 $\{s_n\}$ ...
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.
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 $f\in C[0,1]$ ...
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 ...