11.9 Representations of Functions as Power SeriesTo represent a function as a sum of a power series we manipulate a geometric series or we differentiate or integrate it. Remember from our study of geometric series that 0||1nnaarrr∞==<−1∑. Similarly, this tells us from the perspective of a power series that 01,111nnxrx awhenxx∞====−<−∑1<. So, the function 1( )1f xx=−can be represented as a power series for part of its domain. Look at the graphs below. The graph of the series is on the left and the function graph is on the right. Notice that the graphs are the same.10001nnxforx=−<<∑11( )1f xx=−-1-0.50.5112345-1-0.50.5112345In similar ways other functions can be represented by power series.
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