Alternating SeriesAn alternating series is a series where terms alternate in sign.

1

1

1 1 1 1 1...

1 2 3 4

n

n n

Thm. Alternating Series Test

The alternating series Ʃ(-1)nan will converge if

i)

ii) an is a decreasing sequence (an + 1 < an)

lim 0nna

Ex.

1

1n

n n

Ex. 1

1

1

!

n

n n

Thm. Absolute Convergence Test

If Ʃ|an| converges, then Ʃan converges.

Ex.

31

1n

n n

Ʃan is absolutely convergent if Ʃ|an| and Ʃan

both converge.

Ʃan is conditionally convergent if Ʃan

converges but Ʃ|an| diverges.

3

1

1n

n n

1

1n

n n

Ex. 1

1

1

2 5

n

nn

Ex.

1

1n

n n

Thm. Alternating Series Remainder

If you use N terms to approximate the sum of the convergent alternating series Σ(-1)nan, then the error is less than aN + 1.

Ex. Approximate the value of

using the first 6 terms. How many terms are needed to ensure an error less than .001?

1

1n

n n

Pract.

Determine the convergence, and state the test used

1.

2.

3. Find with an error less than .001

2

1

31

11

n

n

n

n

1

1 3

4 1

n

n

n

n

0

1

!

n

n n

Conv., Alt. Series Test

Div., n-th term test

0.368