How to Evaluate an Improper Integral that Is Vertically Infinite
How to Split One Definite Integral into Two Definite Integrals
Determining Whether a Taylor Series Is Convergent or Divergent

Using the Constant Multiple Rule for Simplifying a Series

The Constant Multiple Rule for integration allows you to simplify an integral by factoring out a constant. This option is also available when you’re working with series. Here’s the rule:

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For example:

image1.png

To see why this rule works, first expand the series so you can see what you’re working with:

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Working with the expanded form, you can factor a 7 from each term:

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Now express the contents of the parentheses in sigma notation:

image4.png

As if by magic, this procedure demonstrates that the two sigma expressions are equal. Of course, this magic is really nothing more exotic than your old friend from grade school, the distributive property.

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