Summary Of Convergence Tests For Series
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Summary of Convergence Tests for Series Test term test (or the zero test) n
Series
X
th
Geometric series
1 X n
=0
or
n
ax
-series
n
=1
n
1 X
1 X
=1
n
ax
1
!
Comparison
an
and
n
Root*
X X
an
!1 bn
with lim
an
with lim
X
bn
a
1 X
!1
X
bn
n
=1
j
an
X
+1 j = L
j nj a
p n j j= n n!1 a
Z1
Z 1c
( ) dx converges
f x
( ) dx diverges
bn
converges =)
an
diverges =)
bn
converges =)
bn
diverges =)
X
X
bn
X
X
an
diverges
an
an
converges
converges
diverges
Converges (absolutely) if L < 1 Diverges if L > 1 or if L is in nite
Diverges if L > 1 or if L is in nite
X (an > 0)
j n j converges =) a
X
an
converges
Converges if 0 < an+1 < an for all n and lim an = 0 n
!1
Useful for comparison tests if the nth term an of a series is similar to axn .
n
f x
c
!1 an = 0.
Useful for comparison tests if the nth term an of 1 a series is similar to p .
Converges (absolutely) if L < 1 L
an
( 1)n 1 an
X
=L>0
n
X
j nj
Alternating series
X
an
Absolute Value
X
Diverges if
with an ; bn > 0 for all n and lim
Ratio
Converges if
with 0 an bn for all n
X
Limit Comparison*
X
and
only if jxj < 1
Diverges if p 1
p
an (c 0) = an = f (n) for all n
an
x
n
Converges if p > 1
n c
X
a
1
Inconclusive if lim
Diverges if jxj 1
1 X
Integral
!1 an 6= 0
Converges to
1 n
Comments
Diverges if lim
an
n
p
Convergence or Divergence
The function f obtained from an = f (n) must be continuous, positive, decreasing and readily integrable for x c. The X comparison series bn is often a geometric series or a p-series. The X comparison series bn is often a geometric series or a p-series. To nd bn consider only the terms of an that have the greatest e ect on the magnitude. Inconclusive if L = 1. Useful if an involves factorials or nth powers. Test is inconclusive if L = 1. Useful if an involves th n powers. Useful for series containing both positive and negative terms. Applicable only to series with alternating terms.
*The Root and Limit Comparison tests are not included in the current textbook used in Calculus classes at Bates College.
1
Series
X
th
Geometric series
1 X n
=0
or
n
ax
-series
n
=1
n
1 X
1 X
=1
n
ax
1
!
Comparison
an
and
n
Root*
X X
an
!1 bn
with lim
an
with lim
X
bn
a
1 X
!1
X
bn
n
=1
j
an
X
+1 j = L
j nj a
p n j j= n n!1 a
Z1
Z 1c
( ) dx converges
f x
( ) dx diverges
bn
converges =)
an
diverges =)
bn
converges =)
bn
diverges =)
X
X
bn
X
X
an
diverges
an
an
converges
converges
diverges
Converges (absolutely) if L < 1 Diverges if L > 1 or if L is in nite
Diverges if L > 1 or if L is in nite
X (an > 0)
j n j converges =) a
X
an
converges
Converges if 0 < an+1 < an for all n and lim an = 0 n
!1
Useful for comparison tests if the nth term an of a series is similar to axn .
n
f x
c
!1 an = 0.
Useful for comparison tests if the nth term an of 1 a series is similar to p .
Converges (absolutely) if L < 1 L
an
( 1)n 1 an
X
=L>0
n
X
j nj
Alternating series
X
an
Absolute Value
X
Diverges if
with an ; bn > 0 for all n and lim
Ratio
Converges if
with 0 an bn for all n
X
Limit Comparison*
X
and
only if jxj < 1
Diverges if p 1
p
an (c 0) = an = f (n) for all n
an
x
n
Converges if p > 1
n c
X
a
1
Inconclusive if lim
Diverges if jxj 1
1 X
Integral
!1 an 6= 0
Converges to
1 n
Comments
Diverges if lim
an
n
p
Convergence or Divergence
The function f obtained from an = f (n) must be continuous, positive, decreasing and readily integrable for x c. The X comparison series bn is often a geometric series or a p-series. The X comparison series bn is often a geometric series or a p-series. To nd bn consider only the terms of an that have the greatest e ect on the magnitude. Inconclusive if L = 1. Useful if an involves factorials or nth powers. Test is inconclusive if L = 1. Useful if an involves th n powers. Useful for series containing both positive and negative terms. Applicable only to series with alternating terms.
*The Root and Limit Comparison tests are not included in the current textbook used in Calculus classes at Bates College.
1
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