Mathematics for Computer Science

15.6. References 661

For example the number 1 may be used to indicate the identity series,I, and 0 may
indicate to the zero series,Z. Whether “r” means the number or the sequence is
supposed to be clear from context.
Verify that in the ring of formal power series,

r ̋.g 0 ;g 1 ;g 2 ;:::/D.rg 0 ;rg 1 ;rg 2 ;:::/:

In particular,
.g 0 ;g 1 ;g 2 ;:::/D 1 ̋.g 0 ;g 1 ;g 2 ;:::/:

Problem 15.25.
Define the formal power series

XWWD.0;1;0;0;:::;0;:::/:

(a)Explain whyXhas no reciprocal.

Hint:What can you say aboutx
.g 0 Cg 1 xCg 2 x^2 C


/?

(b)Use the definition of power series multiplication, ̋, to prove carefully that

X ̋.g 0 ;g 1 ;g 2 ;:::/D.0;g 0 ;g 1 ;g 2 ;:::/:

(c)Recursively defineXnforn 2 Nby

X^0 WWDIWWD.1;0;0;:::;0;:::/;

XnC^1 WWDX ̋Xn:

Verify that the monomialxnrefers to the same power series asXn.

Class Problems

Problem 15.26.
Show that a sequenceGWWD.g 0 ;g 1 ;:::/has a multiplicative inverse in the ring of
formal power series iffg 0 ¤ 0.