Using the Sum Rule for Simplifying a Series

By Mark Zegarelli

The Sum Rule for integration allows you to split a sum inside an integral into the sum of two separate integrals. Similarly, you can break a sum inside a series into the sum of two separate series:

You can break a sum inside a series into the sum of two separate series

For example:

A series in sigma notation.

A little algebra allows you to split this fraction into two terms:

A fraction series expressed in two terms.

Now the rule allows you to split this result into two series:

Using the sum rule to split an integral into two simplified series.

This sum of two series is equivalent to the series that you started with. As with the Sum Rule for integration, expressing a series as a sum of two simpler series tends to make problem-solving easier. Generally speaking, as you proceed onward with series, any trick you can find to simplify a difficult series is a good thing.