AP Calculus
Unit 8 – Infinite Sequences and Series
When encountering a series, there are 10 different tests that you can use. While there is no hard and fast rule,
the tests below are summarized in order of usefulness. When using comparison tests, you should know that
1
1
1
is
divergent,
is
convergent,
and
is divergent.
2
n
n 1 n
n 1 n
n 1
nth term test
lim an
Converge
Diverge
lim an 0
n
Inconclusive
lim an 0
n
n
Special Forms
Name
Geometric
Form
an ar n
Converge
r 1
Telescoping
Alternating
1n an or 1n 1 an
a
a 1 r
n 1
1
an p
n
an bn
p‐series
n
p 1
0 p 1
Subsequent terms cancel
out previous terms.
Converge to terms that
don't cancel out.
an 1 an
No general cancellation
of terms.
Tests
Diverge
Name
Ratio Test
Converge
an 1
1
n a
n
an 1
1
n a
n
Limit Comparison Test
bn is convergent and
a
lim n L (positive and
n b
n
finite)
f n an f is continuous
bn is divergent and
a
lim n L (positive and
n b
n
finite)
Integral Test
lim
and decreasing on a,
Diverge
r 1
an 1 an
Inconclusive
an 1
1
n a
n
lim
lim
f x dx does not exist
a
f x dx L
a
Root Test
Direct Comparison Test
lim n an 1
lim n an 1
lim n an 1
n
n
n
an bn for all n. If bn
converges, then an
converges
an bn for all n. If an
diverges, then bn
diverges
8‐8 Summary of Convergence Tests
AP Calculus
Unit 8 – Infinite Sequences and Series
For each series, determine whether it converges or diverges with a justification. List any other tests that will
give the same conclusion.
n2 4
Use the nth term test.
2
1 to i diverge
n 1 n 2n
In
Lisa
1
n 4
n 1
Looks similar to a p‐series so use the Limit Comparison test with a comparable
2
known p‐series. compare to
1n n 1
The 1 is an indication of the Alternating Series Test, make sure you check for
n 2n
IF
n
2
n 1
It
1
4n
n 1 100 2n
n
conditional/absolute convergence. Limit Comparison test is useful here with a
known p‐series. What would that p‐series be (it's not the same as the example
above).
compare
mm
4n n 4
n 1 n!
n 1
The factorials are an indication to use the Ratio test first.
his
3
i
diverge
How could you rewrite the radical as a power. The 12 and 7 may also be factored
out. This should look like a basic p‐series.
3
need
Factorials again.
converge
12
7 n
i conditionally convergent
divergeby root test
E.tn alwaysdiverge
piansan
The4254273
exponent n is an indication of the root test, however check the nth term test
first.
3n n!
n 1 n 1!
converge by LeT
p
TITTn
2
e
convergent p series
Notice that the base on the power is a collection of constants. What is the value?
n 1
Use the nth term test.
e
n 1 4
I
1
diverge
n
ne
n2
1
This is a geometric series.
r
1 converges
This series is always positive and can be integrated.
n 1
4n 3
use
u
sub
This looks similar to the divergent Harmonic series.
n 1
use L C T with
diverges
Eth
8‐8 Summary of Convergence Tests
AP Calculus
Unit 8 – Infinite Sequences and Series
n 1
n 1 2n 3
n
The exponent n indicates the root test.
converge
4n
n
n
n 1 2 3
sin n
n
n 1 2
n 1
find
Is
since
test
In is a
convergent geometric
series
2h
must converge
by Direct comparison test
since sinn has both
G
and G term
must be
absolutely convergent
since bing.si
t
DNE
diverges by nthterm
In
10
tan n
compareto
divergent harmonic
series
ftp.itamf
1 hospital applies
his sec.cn tk
like to see f
kin
10
E tan
diverges
by L C T
o
8‐8 Summary of Convergence Tests
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