8.6 Alternating Series, Absolute and Conditional
Convergence
n 1
1
an
Alternating series have the form
n 1
a1 – a2 + a3 – a4 + ….
Alternating Series Test
n 1
1
an converges if:
n 1
1. an > 0 for all n
2. an ≥ an+1 for all n ≥ N for some integer N
lim an 0
3. n
Ex.
1
n 1
n 1
1
n
Ex.
1
n 1
n 1
2n 2
4n 2 1
Alternating Series Estimation Theorem
n 1
1
an
If the alternating series
satisfies the three
n 1
conditions of the Alternating Series Test, then for all n ≥ N
n
sn 1 ai a1 a2 a3 ... 1
i1
i1
n 1
an
approximates the sum L of the series with an error whose
absolute value is less than an+1, the value of the first unused
term. The remainder L – sn has the same sign as the first
unused term.
Ex.
1
n 1
n 1
n
n3 1
1. Estimate the error in using the sum of the first four terms to
approximate the sum of the entire series.
2. Approximate the sum so the error has magnitude ≤ .01.
Do: 1. Are the following series convergent or divergent?
1
a.
n 1
n 1
c.
1
n10
n 1
1
d.
n 1
n 1
1
10 n
1
10 n
b.
n 1
1
10 n
1
n10n
n1
e.
f.
1
n1
n1
1
2n 1
2. How many terms of the series do we need to add in order to
find the sum if the error ≤.1?
1
n 1
n 1
1
2n 1
Absolute and Conditional Convergence
n 1
1
an converges absolutely (is absolutely
A series
n 1
convergent) if the corresponding series of absolute values,
an , converges. If
n 1
n 1
an
converges then
1
n 1
n 1
an
converges.
1
1 2
Ex.
n
n 1
n 1
Ex.
1
n 1
n 1
1
3
n
If
n 1
1
an
an
diverges but n 1
series is conditionally convergent.
n 1
converges, then the
0
