Differentiating and integrating a known series can help you find other series. Since we can find the series for cos(x) this way
Of course we could also have integrated the series for sin(x) to get the series for –cos(x) and then changed the signs.
In out next post we will find that
Recall that , so we can integrate the series above to find the series for arctan(x).
Since it follows that the constant of integration so
When differentiating or integrating the interval of convergence cannot get any larger, but it can get smaller. If the endpoints are included before differentiating or integrating then you must check to see if they are included in the new series.
Next post: Rational functions and geometric series.