9.2 Series and Convergence

1. 9.2 Series and Convergence

2. If Sn has a limit as , then the series converges, otherwise it diverges. Series 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

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 seriesconverges to 1. 1

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

9. a r Examples Determine whether the series is convergent or divergent. Example: Write 3.545454… as a rational number.