How to Integrate a Power Series

Because power series resemble polynomials, they’re simple to integrate using a simple three-step process that uses the Sum Rule, Constant Multiple Rule, and Power Rule.

For example, take a look at the following integral:

image0.png

At first glance, this integral of a series may look scary. But to give it a chance to show its softer side, you can expand the series out as follows:

image1.png

Now you can apply the three steps for integrating polynomials to evaluate this integral:

  1. Use the Sum Rule to integrate the series term by term:

    image2.png
  2. Use the Constant Multiple Rule to move each coefficient outside its respective integral:

    image3.png
  3. Use the Power Rule to evaluate each integral:

    image4.png

Notice that this result is another power series, which you can turn back into sigma notation:

image5.png
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