Alternating Series
The last special type of series that
AP requires is alternating series
A series is alternating if every other term is
positive and every other term is negative.
Nth term test for Alternating series: an
alternating series will converge if the nth
term approaches zero as n goes to ∞
Note: if an alternating series converges but the
absolute value of the series diverges then it
is said the to converge conditionally.
Alternating Series
Alternating Series Test
Good news!

example:
  1
n 1
The signs of the terms alternate.
If the absolute values of the terms
approach zero, then an alternating
series will always converge!
n 1
1 1 1 1 1 1 1
       
n 1 2 3 4 5 6
This series converges (by the Alternating Series Test.)
This series is convergent, but not absolutely convergent.
Therefore we say that it is conditionally convergent.

Do the following series converge or diverge?
If they converge is it conditional or absolute
convergence?
Do the following series converge or diverge?
If they converge is it conditional or absolute
convergence?
Since each term of a convergent alternating series
moves the partial sum a little closer to the limit:
Alternating Series Estimation Theorem
For a convergent alternating series, the truncation
error is less than the first missing term, and is the
same sign as that term.
This is a good tool to remember, because it is easier
than the LaGrange Error Bound.
This is typically only used for alternating series that converge absolutely

If a series is conditionally convergent then
it can add to any real number