Sec. 8.4 The Comparison Tests
The Comparison Test: Suppose that n
a
โ and n
b
โ
are series with positive terms.
(a) If n
b
โconverges and nn
abโค for all n, then n
a
โ
converges.
(b) If n
b
โ diverges and nn
abโฅ for all n, then n
a
โ
diverges.
To use the Comparison Test we must use a series to compare for which we already know the convergence or
divergence.
Convergent Series Divergent Series
Geometric Series with |r| < 1 Geometric Series with |r| โฅ 1
Telescoping series like
1
1
(1)
nnn
โ
=+
โ The harmonic series
1
1
nn
โ
=
โ
The series
0
1
!
nn
โ
=
โ Any series n
a
โ
for which lim n
na
โโ does
not exist or lim 0
n
na
โโ
โ
The p-series
0
1
p
nn
โ
=
โ with p > 1 The p-series
0
1
p
nn
โ
=
โ
with p โค 1
EX 1 Use the Comparison Test to find the convergence or divergence of the following series.
A.
()
1
ln 2
k
k
k
โ
=
+
โ
B. 3
1
ln
k
k
k
โ
=
โ
C. 3
1
5
nnn
โ
=+
โ
