8.3 IntegralTest

8.3 IntegralTest - R e v ie w 

 S e r ie s 
 ∞ 1 H a r m o n ic 
 s e r ie s:
 ∑ n 
 n =1 


diverges ∞ ∑ ar n −1

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Unformatted text preview: R e v ie w 

 S e r ie s 
 ∞ 1 H a r m o n ic 
 s e r ie s :
 ∑ n 
 n =1 


diverges. ∞ ∑ ar n −1 Geometric:

 n =1 ∞ or ar n ∑ n =0 
 a if r < 1 .
 1.
A
geometric
series
converges
to
 1 − r The
series
diverges
if
|r|
≥
1.
 
 ∞ ∑a 2.
Telescoping:
 
 
 
 
 
 
 
 
 
 
 
 
 n =1 n − an + c 
 Tests
 
 ∞ 1.
 ∑a n =1 n 
converges
to
L
if
the
sequence
of
its
partial
sums
{Sn}
 converges
to
L.

 ( lim S = L) 
 n→∞ n 
 th 2.
The
n 
Term
Test
for
Divergence
 ∞ ∑a n =1 n
 l diverges
if
 nim an 
fails
to
exist
or
is
different
 →∞ l from
zero.

(
If
 nim an = 0 ,
the
test
is
inconclusive.)
 →∞ 
 ∞ Ex.


Let
an
=
2.

Does
{an}
converge
or
diverge?


Does
 converge
or
diverge?
 
 
 
 
 
 
 ∑a n =1 n 
 Ex.
Let
 an = ∞ ∑a n =1 n n−2 n .
Does
{an}
converge
or
diverge?


Does
 
converge
or
diverge?
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Let
 
 
 
 
 
 
 
 
 
 
 
 ∞ Sn = ∑ n−2 an Does
 n .






 n =1 
converge
or
diverge?
 8.3
Integral
Test
 
 Sometimes
we
can’t
find
the
exact
sum
of
a
series,
but
we
can
 determine
whether
it’s
convergent
or
divergent.
 
 ∞ 1 ∑ n2 Ex.
 n =1 
 
 Out[40]//TableForm= n 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. Sn 0. 1.63498 1.63995 1.64161 1.64244 1.64294 1.64327 1.64351 1.64368 1.64382 1.64393 
 
 ∞ ∑a n Pg.
523:
A
series
 n =1 
of
nonnegative
terms
converges
if
and
 only
if
its
partial
sums
are
bounded
from
above.
 
 2.5 2.0 1.5 1.0 0.5 0.0 
 
 1 2 3 4 5 
 The
Integral
Test
 Let
{an}
be
a
sequence
of
positive
terms.

Suppose
that
an
=
 f(n),
where
f
is
a
continuous,
positive,
decreasing
function
of
x
 ∞ ∑a for
all
x
≥
N
(N
a
positive
integer).

Then
the
series
 n = N ∫ the
integral

 ∞ N 
 ∞ arctan n ∑ 1 + n2 Ex.
 n =1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f ( x ) dx n 

and
 
both
converge
or
both
diverge.
 ∞ ∑ Ex.
 n =1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 n
 ∞ 1 ∑ np Def:

A
p‐series
is
a
series
of
the
form
 n =1 
where
p
is
a
real
 constant.


If
p
≤
1,
the
series
diverges.

If
p
>
1,
the
series
 converges.
 
 ∞ Ex.
 
 ∑ n =1 ∞ Ex.
 
 ∑ n =1 −8 2 5 n
 −8 5 2 n
 ∞ ∑ Do:

Use
the
integral
test
to
determine
if

 n =1 
converges
or
diverges.
 
 
 
 
 
 
 1 n 26 
 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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