1. MTH1170 Series

2. Preliminary
If we start with a sequence and add up all of the terms as n approaches infinity, we will have a series.

3. Notation
Series are represented using sigma notation: This notation says to add up all of the a terms from n = 1 to n = infinity.

4. Partial Sum
A partial sum can be defined using the following notation: Sn is the sum of the first n terms of the sequence a. Notice that every time that n increases we will generate a new value for Sn.

5. Sequence of Partial Sums
If we collect all of the Sn values and order them in a list, we will obtaiIf we collect all of the Sn values and order them in a list, we will obtain a sequence of partial sums:n a sequence of partial sums:

6. Convergence of Series
If the sequence of partial sums {Sn} converges then this means that the limit as n goes to infinity of Sn must also exist.

7. Convergence of Series
If this is true, then the series must also converge and the sum will be finite. S represents the final sum. If {Sn} diverges then the series also diverges.

8. How to Solve Series
Re-write the series as the limit as n goes to infinity of a partial sum.
2. Write out a general expression for the nth term in the sequence of partial sums.
3. Find a formula for Sn.
4. Take the limit of Sn as n goes to infinity. If the limit exists then this is the solution. Otherwise the series diverges.

9. Example
Solve the following series:

10. Geometric Series
A geometric series is a special kind of series that takes the following form:

11. Geometric Series
Geometric series converge for: -1 < r < 1 The solution to a geometric series is 1/1-r.

12. Example
Solve the following series:

13. Example This is a geometric series with r = -1/2.
We know that this series converges.