Summary of Tests for Series - Larson
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Transcript of Summary of Tests for Series - Larson
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646 Chapter 9 Infinite Series
SUMMARY OF TESTS FOR SERIES
Test
Term
Geometric Series
Telescoping Series
Series
Alternating Series
Integral( is continuous,positive, anddecreasing)
Root
Ratio
Direct Comparison
Limit Comparisonan, bn > 0
an, bn > 0
f
p-
nth-
Series
n1an
n1an
n1an
n1an
an f n 0
n1an,
n11n1an
n1
1n p
n1bn bn1
n0arn
n1an
Condition(s)of Convergence
and
converges
and converges
and converges
n1bn
limn
anbn
L > 0
n1bn
0 < an bn
limn an1an < 1lim
nnan < 1
1
f x dx
limn
an 00 < an1 an
p > 1
limn
bn L
r < 1
Comment
This test cannot be usedto show convergence.
Sum:
Sum:
Remainder:
Remainder:
Test is inconclusive if
Test is inconclusive if
limn an1an 1.lim
nnan 1.
0 < RN < N
f x dx
RN aN1
S b1 L
S a1 r
Condition(s)of Divergence
diverges
or
or
and diverges
and diverges
n1bn
limn
anbn
L > 0
n1bn
0 < bn an
limn an1an > 1
limn
nan > 1
1
f x dx
0 < p 1
r 1
limn
an 0