1. Special Series (Part 2)
Dr. Shildneck

2. Algebra Skill – Partial Fractions
The major skill required for evaluating the series we will discuss today is decomposing a rational function into the sum of two smaller rational functions, called partial fractions.
Instead of taking two fractions and putting them together,
We want to take the fraction that is already put together and break it apart.

3. Algebra Skill – Partial Fractions
In order to do this (for the purposes we will investigate) proceed as follows:
Factor the denominator.
Write the expression as the sum of two fractions where each factor is the denominator of one of the fractions. Make the unknown numerators A and B.
Add the fractions by getting a common denominator.
Set the numerator of the new fraction equal to the original numerator.
Write a system of equations in terms of A and B.
Solve for A and B.
Rewrite the decomposition using the values for A and B.

4. Algebra Skill – Partial Fractions
Factor the denominator.
Write the expression as the sum of two fractions where each factor is the denominator of one of the fractions. Make the unknown numerators A and B.
Add the fractions by getting a common denominator.
Set the numerator of the new fraction equal to the original numerator.
Write a system of equations in terms of A and B.
Solve for A and B.
Rewrite the decomposition using the values for A and B.
[Example 1]

5. Algebra Skill – Partial Fractions
Factor the denominator.
Write the expression as the sum of two fractions where each factor is the denominator of one of the fractions. Make the unknown numerators A and B.
Add the fractions by getting a common denominator.
Set the numerator of the new fraction equal to the original numerator.
Write a system of equations in terms of A and B.
Solve for A and B.
Rewrite the decomposition using the values for A and B.
[Example 2]

6. Telescoping Series
A telescoping series is a series that can be expanded (like the layers in a telescope) using partial fractions.
When expanded, telescoping series will typically have a pattern to them that will simplify.
For FINITE telescoping series, this might make our summation much shorter.
INFINITE telescoping series are similar to infinite geometric series. That is, they will either converge to a single value (which you can find) or diverge and increase/decrease infinitely without a “final” value

7. Telescoping Series [Example 3] Decompose the rational function
Begin expanding the series to determine the pattern.
Complete enough iterations that you can determine how the pattern simplifies.
Simplify as much as possible.
Evaluate.
[Example 3]

8. Infinite Telescoping Series
Decompose the rational function
Begin expanding the series to determine the pattern.
Complete enough iterations that you can determine how the pattern simplifies.
Simplify as much as possible. For infinite series, determine if the series converges or diverges.
Evaluate.
[Example 4]

9. Assignment
Worksheet 2 – Special Series Part 2 (Telescoping Series)