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9-2 Series. AP Calculus Miss Battaglia. Summing Series. An 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:.

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9 2 series

9-2 Series

AP Calculus

Miss Battaglia


Summing series
Summing Series

  • An 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:



Partial sums
Partial Sums 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.


The main event convergence and divergence of a series
The Main Event: Convergence and Divergence of a Series more compact form:

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.


A no brainer divergence test the nth term test
A no-brainer divergence test: more compact form: The nth term test


Geometric series
Geometric Series more compact form:

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 rule
Geometric Series Rule more compact form:

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

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

diverges.

Ex: a = 5 and r = 3



Telescoping series
Telescoping Series more compact form:

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

A telescoping series will converge iffbn approaches a finite number as n∞. Moreover, if the series converges it sum is


Writing a series in telescoping form
Writing a Series in Telescoping Form more compact form:

Find the sum of the series


Homework
Homework more compact form:

  • P. 614 #9, 12, 17, 18, 37, 39, 43, 45, 51, 59, 61, 68, 69, 70, 71, 72, 74


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