Page "Formal power series" Paragraph 43

Informally, two sequences ( a < sub > n </ sub >) and ( b < sub > n </ sub >) become closer and closer if and only if more and more of their terms agree exactly.

Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.

This is clearly the case for the right hand side of ( 1 ), regardless of the values a < sub > n </ sub >, since inclusion of the term for i = n gives the last ( and in fact only ) change to the coefficient of X < sup > n </ sup >.

It is also obvious that the limit of the sequence of partial sums is equal to the left hand side.