By Mark Zegarelli

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:

The integral of a power series.

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:

An expanded power series

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:

    Using the sum rule to integrate the terms of a series.

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

    The constant multiple rule allows you to move the coefficients outside of the integrals.

  3. Use the Power Rule to evaluate each integral:

    Using the power rule to evaluate each integral

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

An integral in the sigma notation.