New posts in sequences-and-series

The "Largeness" of Subsets of the Natural Numbers

sequences-and-series number-theory order-theory

Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges

real-analysis sequences-and-series convergence-divergence prime-numbers analytic-number-theory

Sequence $a_n = a_{n - 1} + a_{\lfloor n/2 \rfloor}$

sequences-and-series number-theory

How to prove that $\sum_{k=1}^{\infty}\frac{k^{n+1}}{k!}=eB_{n+1}=1+\cfrac{2^n+\cfrac{3^n+\cfrac{4^n+\cfrac{\vdots}{4}}{3}}{2}}{1}$

sequences-and-series continued-fractions

Is there convenient notation for Viète's formula?

sequences-and-series notation

Irrationality of Two Series

number-theory sequences-and-series irrational-numbers

Convergence of a product series with one divergent factor

real-analysis sequences-and-series convergence-divergence

sum of series involving coth using complex analysis

sequences-and-series complex-analysis trigonometry

Does Newton's method for inverting a series work?

calculus real-analysis sequences-and-series convergence-divergence

Infinite Series $\sum\limits_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$

calculus sequences-and-series analysis

Problem with infinite product using iterating of a function: $ \exp(x) = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x) \cdot \ldots $

sequences-and-series number-theory functional-analysis

How will the limit $\lim \sin(A^{n})$ behave for $|A| > 1$?

sequences-and-series limits convergence-divergence

The space of sequences as a complete metric space

sequences-and-series functional-analysis metric-spaces

$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent

real-analysis sequences-and-series convergence-divergence continuity

Recursive Sequence $a_n = \frac{1}{2} (a_{n-1} + 5) $

real-analysis sequences-and-series limits recurrence-relations

Prove that $inf\ \{|x_n|, n \in \mathbb{N}\}=0$

sequences-and-series solution-verification supremum-and-infimum

Does the sequence $f_1=x^2+1$ , $f_{n+1}=(f_n)^2+1$ contain only irreducible polynomials?

sequences-and-series polynomials dynamical-systems irreducible-polynomials

What is the closed-form for $\displaystyle\sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + mn+41n^2}$?

sequences-and-series algebraic-number-theory radicals zeta-functions

Show that $\forall n \in\Bbb N: e < \left(1+{1\over n}\right)^n \left(1 + {1\over 2n}\right)$

calculus sequences-and-series limits proof-verification inequality

A peculiar Euler sum

sequences-and-series special-functions polylogarithm summation-by-parts