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Series and Convergence
9.2 Series and Convergence
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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.
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Examples Determine whether the series is convergent or divergent.
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Divergence Test If then the series diverges.
Examples: Determine whether the series is convergent or divergent. If it is convergent, find its sum.
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Example Using partial fractions:
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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:
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Example Determine whether the series is convergent or divergent. 1
This infinite series converges to 1. 1
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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.
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Examples Determine whether the series is convergent or divergent. a r
Example: Write … as a rational number.
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