CONVERGENCE TESTS FOR INFINITE SERIES
NAME
COMMENTS
STATEMENT
Geometric series
!
ar
k
=
a
1 – r
, if –1 < r < 1
Geometric series converges if
–1 < r < 1
and diverges otherwise
Divergence test
(nth Term test)
If
k
!
!
lim
a
k
"
0, then
!
a
k
diverges.
If
k
!
!
lim
a
k
= 0,
!
a
k
may or may not converge.
p – series
If p is a real constant, the series
!
1
a
p
=
1
1
p
+
1
2
p
+ . . . +
1
n
p
+ . . .
converges if p > 1 and diverges if 0 < p
#
1.
Integral test
!
a
k
has positive terms, let f(x) be a function that results when k is
replace by x in the formula for u
k
.
If is decreasing and continuous for
x
$
1, then
!
a
k
and
!
"
1
%
f(x) dx
both converge or both diverge.
Use this test when f(x) is easy to integrate.
This
test only applies to series with positive terms.
Comparison test (Direct)
If
!
a
k
and
!
b
k
are series with positive terms such that each term in
!
a
k
is less than its corresponding term in
!
b
k
, then
(a) if the "bigger series"
!
b
k
converges, then the "smaller

This is the end of the preview. Sign up
to
access the rest of the document.