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:
To see why this rule works, first expand the series so you can see what you’re working with:
Working with the expanded form, you can factor a 7 from each term:
Now express the contents of the parentheses in sigma notation:
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.