Summation Properties

Let's use examples to derive some summation properties. First recall that where ak stands for the kth term of the sequence. As you read and work through these examples, you should notice that they all just rely on algebraic and arithmetic properties that you are already familiar with. The only thing new here is the application of these properties to the summation idea. Thus once you understand what that little summation symbol stands for, you should be able to derive any of these properties as you need them. Also, you will be asked some problems where you need a combination of the following ideas or maybe even a similar idea that is not explicitly listed here. Make sure you ask about anything that is confusing to you.

 

1. Sum of a constant:

Example 1: Evaluate and and use your discovery to evaluate:

 

 

Fill in the general property:

 

 

2. A constant multiplied by each term of the sequence:

Example 2: Suppose . Evaluate

 

 

Fill in the general property:

Note that although we used a finite sum in this example, the property would also work for infinite sums:

 

 

3. Break up a summation:

Example 3: Given: and and , evaluate .

 

 

Fill in the missing parts of the general property:

where m < n and m, n are positive integers.

 

 

4. Adjust the indices of a summation:

Example 4: Given that: , evaluate

 

 

 

Fill in the missing parts of the general property:

 

 

5. Break up the argument of the summation when there is more than one term:

Example 5: Evaluate if we know:

 

 

 

In general:

 

 

 

6. Pairing of terms:

In some special cases, you can pair off terms and use this pairing to find the sum. (This is also an arithmetic series.):

Find a shortcut to evaluating:

 

 

In general:

 

Of course you could make up all kinds of problems that can use one or several of these properties. See the summation problem set.

Warning: . Can you figure out why?