A.9. SERIES CONVERGENCE TESTS 173
A.9 SERIES CONVERGENCE TESTS
DIVERGENCE TEST
If lim
n!1
x
n
¤ 0 then
1
X
nD0
x
n
does not have a finite value.
INTEGRAL TEST
Suppose that f .x/ is a positive, decreasing function on Œ0; 1/ and that x
n
D f .n/. en
1
X
nD1
x
n
and
Z
1
a
f .x/dx both converge or both diverge for any finite a 1.
P-SERIES TEST
e p-series
1
X
nD1
1
n
p
converges if p > 1 and diverges if p 1.
COMPARISON TESTS
• If
1
X
nD0
converges and 0 y
n
x
n
for all n, then
1
X
nD0
y
n
also converges.
• If
1
X
nD0
x
n
diverges and y
n
x
n
0 for all n, then
1
X
nD0
y
n
also diverges.
LIMIT COMPARISON TEST
Suppose that
1
X
nD0
x
n
and
1
X
nD0
y
n
are series, and lim
n!1
x
n
y
n
D C where 0 < jC j < 1 is a constant.
en, either both series converge or both series diverge.
ABSOLUTE CONVERGENCE TEST
If
1
X
nD0
jx
n
j converges, then so does
1
X
nD0
x
n
.
ALTERNATING SERIES TEST
Suppose that x
n
is a series such that x
n
> x
nC1
0, and suppose that lim
n!1
x
n
D 0. en the
series
1
X
nD0
.1/
n
x
n
converges.
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