8.4 Comparison Test

8.4 Comparison Test - 8.4
Comparison
Tests



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Unformatted text preview: 8.4
Comparison
Tests

 Recall
we
proved
that
the
Harmonic
Series
diverged
because
 ∞ ∞ 1 1 ∑ n >1+ ∑ 2 n =1 n =1 
 
 Comparison
Test
 ∑ a be
a
series
with
nonnegative
terms.
 a.
 ∑ a converges
if
there
is
a
convergent
series ∑ c Let
 n
 n
 
 with


an
≤
cn
for
all
n
>
N
for
some
integer
N.
 n
 ∑ a diverges
if
there
is
a
divergent
series
of
nonnegative
 terms ∑ d with


d 
≤
a 
for
all
n
>
N
for
some
integer
N.
 b.
 n
 
 n
 n 
 
 ∞ sin n Ex.
 ∑ n 2 n =1 
 
 
 
 
 
 
 
 
 
 n ∞ Ex.
 
 
 
 
 ∑ n =10 1 n − 3
 ∞ 5 Ex.
 ∑ 2 + 3n n =1 
 
 
 
 
 
 
 
 Limit
Comparison
Test
 Suppose
that
an
>
0
and
bn
>
0
for
all
n
≥
N
(N
an
integer)
 lim 1. If
 n → ∞ an =c>0 
then
 ∑ an 
and
 ∑ bn 

both
converge
 bn or
both
diverge.

 lim 2. If

 n → ∞ an =0 
and
 ∑ bn converges,
then

 ∑ an bn converges.
 lim 3. 
If

 n → ∞ an =∞ 
and
 ∑ bn diverges,
then

 ∑ an 
 bn diverges.


























































































 n+2 Ex.
 ∑ n 2 − 5 n + 1 n =1 
 ∞ 
 
 
 
 
 
 
 
 n+2 Ex.
 ∑ 5 n 3 + 3 n =1 
 ∞ 
 
 
 
 
 
 
 
 ∞ 1 Ex.
 ∑ 2 n − 1 n =1 
 
 
 
 
 
 
 
 
 
 ∞ n3 Ex.
 ∑ 5 n 2 + 1 n =1 
 
 
 ∞ Ex.
 
 
 
 ∑ n =1 1 n − 3
 Do:

Do
the
following
series
converge
or
diverge?
 ∞ ∑ 1.
 n =1 2 n 3n + 1 
 
 ∞ ∑ 2.
 n =1 3 n2 n 
 
 3 + cos n ∑ 3n 3.
 n =1 
 ∞ 
 
 3 + 2n ∑ 3n 4.
 n =1 
 ∞ 
 n n+2 − ∑2 ln n ln(n + 2) 5.
 n = 
 ∞ 
 ∞ 1 n ⎛ 1⎞ ∑⎝ n⎠ 
 6.
 n =1 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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