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

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