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 much
easier than the LaGrange Error Bound (learned later).
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
Homework: 11-35 odd 41,43,51-61 odd
There is a flow chart on page 505 that might be helpful
for deciding in what order to do which test. Mostly this
just takes practice.
To do summations on the TI-89:
1

8 

2
n1
5
n
becomes
 8*(1/ 2 ^ n, n,1,5)
F3

1

8 

2
n 1
n
becomes
31
4
4
 8*(1/ 2 ^ n, n,1, )
8

To graph the partial sums, we can use sequence mode.
MODE
Y=
Graph…….
4
ENTER
u1   (8*(3/ 4) ^ k , k ,1, n)
ENTER
WINDOW
GRAPH

To graph the partial sums, we can use sequence mode.
Graph…….
MODE
Y=
4
ENTER
u1   (8*(3/ 4) ^ k , k ,1, n)
ENTER
WINDOW
GRAPH
Table

To graph the partial sums, we can use sequence mode.
Graph…….
MODE
Y=
4
ENTER
u1   (8*(3/ 4) ^ k , k ,1, n)
ENTER
WINDOW
GRAPH
Table
p
Absolute Convergence
If
a
n
converges, then we say
a
n
converges absolutely.
The term “converges absolutely” means that the series
an converges,
formed
the absolute
value aof
term
If by taking
then
converges.
n each
converges. Sometimes in the English language we use
the word “absolutely” to mean “really” or “actually”. This is
If
the
series
not
the
caseformed
here! by taking the absolute value of each
term converges, then the original series must also
converge.


“If a series converges absolutely, then it converges.”

Tests we know so far:
Try this test first
nth term test (for divergence only)
Then try these
Special series:
Geometric, Alternating, P series, Telescoping
General tests:
Direct comparison test,
Limit comparison test,
Integral test,
Absolute convergence test (to be used with another test)
Homework
p.639 11-33 odd, 51 -67 odd 87-95 odd