Power Series as Representations of Functions

We have seen above that, for a geometric progression of the type given in equation (1.10), the sum of the first n terms is given by equation (1.20). Furthermore, for a = 1, we can see that  

This is an important expression because it allows us to see how a function such as can be represented by a polynomial of degree n - 

However, if we now extend the progressionindefinitely to form the infinite geometric series 1 + x + x + + x + , we obtain an expansion of a function lim which converges only for values of x in the range 

The infinite geometric series is an example of a power series because it contains a sum of terms involving a systematic pattern of change in the power of. X. In general, the simplest form of a power series is given by  

Power series are so-called because they are sums of powers of jf with specified coefficierits. 

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