Example 8.4.18 (d): Finding Taylor Series by Integration
Start with a known series and integrate both sides.
Example
Which function is represented by the series 1/n xnOur known series with which to start is, once again, the Geometric series. For variety, let's use t as variable:
1/1-t = tn = tn-1
Integrating both sides gives:
1/1-t dt = tn-1 dt = tn-1 dt = 1/n xn
Thus, the function represented by this series is:
1/n xn = 1/1-t dt = -ln(1-x)
1/n xn f(x) = -ln(1-x)