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MTH 122 — Calculus II
Essex County College — Division of Mathematics and Physics
1
Lecture Notes #17 — Sakai Web Project Material
1
Introduction to Sequences and Series, Part IV
1.
Comparison Test
: Suppose that
∞
X
n
=1
a
n
and
∞
X
n
=1
b
n
are series with positive terms.
(a) If
∞
X
n
=1
b
n
is convergent, and 0
< a
n
≤
b
n
for all
n
, then
∞
X
n
=1
a
n
is also convergent.
(b) If
∞
X
n
=1
b
n
is divergent, and
a
n
≥
b
n
>
0 for all
n
, then
∞
X
n
=1
a
n
is also divergent.
Example:
Use the comparison test to determine whether
∞
X
n
=1
1
n
3
+ 1
converges.
Work:
For
n
≥
1 we know that
n
3
≤
n
3
+ 1, so
0
<
1
n
3
+ 1
≤
1
n
3
.
You should note that
∞
X
n
=1
1
n
3
is a convergent
p
series. The conclusion, using the
Comparison Test
, is that
∞
X
n
=1
1
n
3
+ 1
also
converges
.
The two big series that you should use for comparisons are the
p
series and the geometric
series. The harmonic series is a
p
series, with
p
= 1.
The Geometric Series
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 Spring '10
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 Calculus, Division, Sequences And Series

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