New posts in power-series

Solutions in terms of the hypergeometric functions

complex-analysis ordinary-differential-equations power-series special-functions hypergeometric-function

What is the radius of convergence of $\sum z^{n!}$?

complex-analysis convergence-divergence power-series

How to express $(1+x+x^2+\cdots+x^m)^n$ as a power series?

real-analysis power-series

How solve this nonlinear trigonometric differential equation

ordinary-differential-equations power-series special-functions trigonometric-integrals nonlinear-analysis

Are the limits $\lim\limits_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|\,$ and $\lim\limits_{n\to \infty }\sqrt[n]{|a_n|}\,$ equal?

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

Necessary and sufficient conditions for a polynomial in $\mathbb{Z}[t]$ to have an $n$th root in $\mathbb{Z}[[t]]$

abstract-algebra power-series

Multiplicative inverses of formal series with non-negative coefficients

sequences-and-series power-series

Taylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?

calculus real-analysis power-series taylor-expansion

Existence of a power series converging non-uniformly to a continuous function

convergence-divergence continuity power-series

Is there a generalization of the fundamental theorem of algebra for power series?

calculus real-analysis power-series

Singularities of $e^{z - \frac{1}{z}}$

complex-analysis power-series singularity-theory

Prove that a power series that is zero on a sequence that converges to zero is the zero function

real-analysis sequences-and-series power-series

Riemann Zeta Function Manipulation

complex-analysis power-series riemann-zeta analyticity

continuity of power series

calculus real-analysis analysis power-series

Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

sequences-and-series power-series pi transcendental-numbers constants

Can $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$ be written as a power series in a neighbourhood of zero?

power-series analytic-functions

Behavior of the roots of an infinite series.

complex-analysis polynomials complex-numbers power-series

Does $\sum_{i=0}^\infty{i\left((1-p^{i+1})^m-(1-p^{i})^m\right)}$ go to infinity as $\log m$?

asymptotics power-series

Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$

power-series riemann-zeta infinite-product dirichlet-series

Why is $ \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?

sequences-and-series algebra-precalculus power-series exponential-function