1. Series and Convergence
9.2
Series and Convergence

2. Series a1, a2,… are terms of the series. an is the nth term.
If we add all the terms of a sequence, we get a series:
a1, a2,… are terms of the series.
an is the nth term.
To find the sum of a series, we need to consider the partial sums:
nth partial sum
If Sn has a limit as , then the series converges, otherwise it diverges.

3. Examples
Determine whether the series is convergent or divergent.

4. Divergence Test If then the series diverges.
Examples: Determine whether the series is convergent or divergent. If it is convergent, find its sum.

5. Example
Using partial fractions:

6. Telescoping Series
A telescoping series is any series that can be written in the following (or similar) form
in which nearly every term cancels with a preceding or following term. However, it doesn’t have a set form.
Partial fraction decomposition is often used to put in the above form.
Partial sum will be considered since most terms can be canceled.
Example:

7. Example Determine whether the series is convergent or divergent. 1
This infinite series converges to 1. 1

8. Geometric Series
In a geometric series, each term is found by multiplying the preceding term by the same number, r.
This converges to if , and diverges if
is the interval of convergence.

9. Examples Determine whether the series is convergent or divergent. a r
Example: Write … as a rational number.