Need help to understand concept of power series

Question

I have done questions using basic ratio test and root test without even being aware of $\lim $as not ordinary but $\limsup$ . I do not know what is relation of $\limsup$ to power series is and how it came up. The only thing I know about $\limsup$ is that it is highest of all set of limit points of a set. Can anybody suggest some reference or explain this.

Answer 1

When you first meet series the individual terms of a series are just numbers, as in $$1-{1\over2}+{1\over3}-{1\over4}+\ldots,\quad\sum_{k=0}^\infty 2^{-k},\quad \sum_{k=2}^\infty {1\over\log k},\quad \sum_{k=1}^\infty{1\over k^2}\ .$$ Given such a series, one can test whether it converges, and if yes, find out whether the sum can be written in a simple way.

In order to construct new interesting functions one then considers function series, i.e. series of the form $$\sum_{k=0}^\infty f_k(x)\ ,\tag{1}$$ where the individual terms are functions with a common domain, say, an interval $D\subset{\mathbb R}$. Plugging in an $x\in D$ into $(1)$ one obtains a "constant" series of the type considered at the beginning, which may or may not converge. Therefore $(1)$ possesses a domain of convergence $D'\subset D$ consisting of all $x\in D$ for which $(1)$ converges.

Now a particular important type of function series are the power series $$\sum_{k=0}^\infty a_k\>x^k\ ,\tag{2}$$ where the coefficient sequence ${\bf a}:=(a_k)_{k\geq0}$ is a completely arbitrary sequence of complex numbers. This sequence ${\bf a}$ can be viewed as the data vector encoding the series $(2)$. Therefore it should be possible to extract all information about the behavior of $(2)$ out of this data vector. In particular the domain of convergence $D'$ of the series $(2)$ is completely determined by ${\bf a}$. It turns out that $D'$ is an interval of the form $\ ]{-\rho},\rho[\ $ (respectively a disk $D_\rho\subset{\mathbb C}$). When the $a_n$ go to zero very fast as $n\to\infty$ this fosters the convergence of $(2)$, and the convergence radius $\rho$ will be large. On the other hand, if there is a single subsequence of $a_n$'s that grows exponentially this has a devastating effect on $\rho$. A careful worst case analysis shows that $\rho$ is given by the formula $$\rho={1\over\lim\sup_{n\to\infty}\bigl|a_n\bigr|^{1/n}}\quad\in[0,\infty]\ .$$ For the definition of $\lim\sup$ see here:

http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior