11.8 Power Series

Recall that we were able to analyze all geometric series "simultaneously''to discover that$$\sum_{n=0}^\infty kx^n = {k\over 1-x},$$if $|x|< 1$, and that the series diverges when $|x|\ge 1$. At the time,we thought of $x$ as an unspecified constant, but we could just aswell think of it as a variable, in which case the series$$\sum_{n=0}^\infty kx^n$$is a function, namely, the function $k/(1-x)$, as long as$|x|< 1$. While $k/(1-x)$ is a reasonably easy function to deal with,the more complicated $\sum kx^n$ does have itsattractions: it appears to be an infinite version of one of thesimplest function types—a polynomial. This leads naturally to thequestions: Do other functions have representations as series? Is therean advantage to viewing them in this way?The geometric series has a special feature that makes it unlike atypical polynomial—the coefficients of the powers of $x$ are thesame, namely $k$. We will need to allow more general coefficients ifwe are to get anything other than the geo...