# Study mathematics by doing it

## September 16, 2011

### Independent University of Moscow. Algebra-1. Ex. sheet №2

Filed under: math — liberium @ 08:06

Problem 1. Find formula for sum of two formal series.

Solution. $\sum_{n=0}^{\infty}{a_n t^n} + \sum_{n=0}^{\infty}{b_n t^n} = \sum_{n=0}^{\infty}{(a_n+b_n) t^n}$

Problem 2. Find formula for product of two formal series.

Solution. $\sum_{n=0}^{\infty}{a_n t^n} \times \sum_{n=0}^{\infty}{b_n t^n}=\sum_{n=0}^{\infty}{ (\sum_{k=0}^{n}{a_k b_{n-k} })t^n }$

Theorem 1. Series $\sum_{n=0}^{\infty}{a_n t^n}$ is invertible iff $a_0 \neq 0$

Proof. Indeed, if $a_0=0$, then series with $a_i=0, i \in \mathbb{N}$ evidently has no inverse. Conversely if $a_0 \neq 0$, then an inverse of the series is $\sum_{n=0}^{\infty}{b_n t^n}$ with $b_0=a_0^{-1},b_n=-(\sum_{k=0}^{n-1}{b_k a_{n-k}}) \cdot a_0^{-1}, n \in \mathbb{N}$ $\Box$