Power Series


Integrate

\begin{displaymath}\frac{d\ln(x+1)}{dx} = \frac{1}{1-(-x)}\end{displaymath}

on both sides and use the geometric series formula on the right to obtain

displaymath145

Performing the integration term by term on the right side you obtain

displaymath146

Plugging in the center tex2html_wrap_inline158 yields tex2html_wrap_inline160 , consequently C=0 and we are done:

displaymath147

By inspection, the center of the power series is tex2html_wrap_inline158 ; you should expect this formula to be valid for |x|<1. Why?


Where to go next?

Continue with the concept of the radius of convergence of a power series next!


Helmut Knaust
Wed Jul 10 13:02:07 MDT 1996
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