# 9 2 Series and Convergence Series If we

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9. 2 Series and Convergence

Series If we add all the terms of a sequence, we get a series: a 1, a 2, … 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 otherwise it diverges. , then the series converges,

Examples Determine whether the series is convergent or divergent.

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

Example Using partial fractions:

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. Ø Partial fraction decomposition is often used to put in the above form. Ø Partial sum will be considered since most terms can be canceled. Example:

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

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. .

Examples Determine whether the series is convergent or divergent. a r Example: Write 3. 545454… as a rational number.