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 The signs of the terms alternate.

Good news!

example: 1

1

1 1 1 1 1 1 11

1 2 3 4 5 6n

n n

This series converges (by the Alternating Series Test.)

If the absolute values of the terms

approach zero, then an alternating

series will always converge!

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:

5

1

18

2

n

n

becomes ^ , ,1,5)8*(1/ 2 n n 31

4

1

18

2

n

n

becomes ^ , ,1, )8*(1/ 2 n n 8

F3 4

To graph the partial sums, we can use sequence mode.

MODE Graph……. 4 ENTER

Y= u1 ( 8*( 3/ 4) ^ , ,1, )k k n WINDOW

ENTER

GRAPH

To graph the partial sums, we can use sequence mode.

MODE Graph……. 4 ENTER

Y=

WINDOW

ENTER

GRAPH

Table

u1 ( 8*( 3/ 4) ^ , ,1, )k k n

To graph the partial sums, we can use sequence mode.

MODE Graph……. 4 ENTER

Y=

WINDOW

ENTER

GRAPH

Table

u1 ( 8*( 3/ 4) ^ , ,1, )k k n

Absolute Convergence

If converges, then we say converges absolutely.na naThe term “converges absolutely” means that the series formed by taking the absolute value of each term converges. Sometimes in the English language we use the word “absolutely” to mean “really” or “actually”. This is not the case here!

If converges, then converges.na na

If the series formed 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 firstnth term test (for divergence only)Then try theseSpecial series: Geometric, Alternating, P series, TelescopingGeneral 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