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9-2 SeriesAP Calculus Miss BattagliaAn infinite series (or just a series for short) is simply adding up the infinite number of terms of a sequence. Consider:

The series associated with the sequence is:Summing Series

You can use fancy summation notation to write this sum in a more compact form:

Continuing with the same series, look at how the sum grows by listing the sum of one term, two terms, three terms, etc.

The nth partial sum, Sn, of an infinite series is the sum of the first n terms of the series.If you list the partial sums, you have a sequence of partial sums.Partial Sums

If the sequence of partial sums converges, you say that the series converges; otherwise, the sequence of partial sums diverges and you say that the series diverges.The Main Event: Convergence and Divergence of a Series

A no-brainer divergence test: The nth term test

A geometric series is a series of the form:

The first term, a, is called the leading term. Each term after the first equals the preceding term multiplied by r, which is called the ratio.

Ex: a = 5 and r = 3

Geometric Series

If 0 < |r| < 1, the geometric series

converges to . If |r| > 1, the series diverges.

Ex: a = 5 and r = 3Geometric Series Rule

Convergent and Divergent Geometric Series

To see that this is a telescoping series you have to use partial fractions.

A telescoping series will converge iff bn approaches a finite number as n. Moreover, if the series converges it sum isTelescoping Series

Find the sum of the seriesWriting a Series in Telescoping Form

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