8.2 Series - R e v ie w 
o f
s e q u e n c e s 
 1...

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Unformatted text preview: R e v ie w 
o f
s e q u e n c e s 
 1 .

{ a n } 
is 
a 
s e q u e n c e 
w h e r e 
a n 
is 
t h e 
n th
t e r m 
o f
t h e 
 s e q u e n c e .
 2 .
A 
s e q u e n c e 
is 
a 
fu n c t io n .
 l 3.
A
sequence
converges
if
 nim an = L 
where
L
is
a
finite
 →∞ r e a l 
n u m b e r .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 .2 
I n fin it e 
S e r ie s 
 An
infinite
series
is
an
infinite
sum.
 ∞ ∑a n n =1 = a1 + a2 + a3 + ... + an + .... 
 
 an
is
the
nth
term
of
the
series
 n Sn = ∑ ai = a1 + a2 + a3 + ... + an i =1 partial
sum.
 
 ∞ Ex.
 
 
 
 
 
 
 ∑n
 n =1 ∞ 1 Ex.
 ∑ n 3 
 n =1 
 
 
 
 
 
is
the
nth
 A
geometric
series
has
the
form
a
+
ar
+
ar2
+
ar3
+
…
+arn
+
….
 ∞ or
 ar n −1 ∑ n =1 ,

where
a
and
r
are
real
numbers,
a
≠
0.
 
 Ex.
2
+
4
+
8
+
16
+
32
+
….
 
 
 
 
 
 
 
 
 
 
 
 ∞ 5 (−1) n 
 Ex.
 ∑ 4 n =1 
 
 
 
 
 
 
 
 
 
 
 n ∞ Def:

The
infinite
series
 ∑a n =1 n 
converges
and
has
sum
“s”
if
the
 sequence
of
partial
sums
{Sn}
converges
to
s.


If
{Sn}
diverges,
 then
the
series
diverges.
 
 a if r < 1.
 A
geometric
series
converges
to
 1 − r The
series
diverges
if
|r|
≥
1.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ∞ 2(2 ) Ex.
 ∑ 
 n −1 n =1 
 
 
 
 ∞ (−1) n Ex.
 ∑ n =1 5 4n 
 
 
 
 
 
 
 
 
 Do:

Write
out
the
first
few
terms
of
the
series.

Find
a
and
r.

 Then
find
the
sum.
 ∞ ∑ 
 
 
 
 
 
 
 
 n =1 n ⎛ 1⎞ 3 ⎝ 4⎠ 
 Find
the
values
of
x
for
which
the
series
converges.

Find
the
 sum
of
the
series
for
those
values
of
x.
 
 ∞ ∑ 
 
 
 
 
 
 
 
 
 
 
 
 
 n =1 ∞ ∑ 
 
 
 
 
 
 
 
 
 n =0 (−1) n x n 
 ( x + 3) n 2n 
 Telescoping
series
 
 ∞ Ex.
 
 
 
 
 
 
 
 
 
 
 
 ∑ n =1 ∞ Ex.
 
 
 
 
 
 
 
 
 
 
 
 
 ∑ n =1 1 1 − n +1 n + 3
 1 1 − ln( n + 2) ln( n + 1) 
 Tests
for
convergence:
 l 1. Find
an
explicit
formula
for
{sn},
then
look
at
 nim →∞ ∞ Ex.
 ∑n
 n =1 
 
 
 
 
 
 
 
 
 ∞ 1 Ex.
 ∑ n 
 n =1 
 
 
 
 
 
 Sn 
 
 2. The
nth
Term
Test
for
Divergence
 ∞ ∑a n =1 n 
diverges
if
 lim an 
fails
to
exist
or
is
different
 n→∞ fr o m 
z e r o .
 
 ∞ ∑a Note:

If
 n =1 n 
converges,
then
 lim an = 0 .

However
if
 n→∞ lim an = 0 ,
the
series
does
not
necessarily
converge.
 n→∞ Y o u 
m u s t 
u s e 
a n o t h e r 
t e s t .
 
 ∞ 1 1 lim = 0 

but
 
diverges.
 n→∞ n n =1 n ∑ 
 ∞ n2 Ex.
 ∑ 2 n 2 + 1 
 n =1 
 
 
 
 ∞ ⎛n⎞ ln Ex.
 ∑ ⎝ 2 n + 1⎠ 
 n =1 
 
 
 cos( nπ ) Ex.
 ∑ 5n 
 n =0 ∞ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Rules:
 ∞ 1.
 2.
 ∞ ∑ ka n =1 ∞ ∑a n =1 3.

If
 = k ∑ an 
 n n n =1 ∞ ∞ n =1 n =1 ∞ ± bn = ∑ an ± ∑ bn 
 ∞ ∑a n =1 ∞ n 
diverges
then
 4.


If
 n 
and
 converges.
 n =1 n 
both
converge,
then
 n 
converges
and
 diverges.
 
 ∞ If
 
 n 
and
 ∑b n =1 ∞ Ex.
 
 
 ∑n − n
 n =1 ∞ Ex.
 ∑ n + n
 n =1 n =1 ∑b n =1 n n ± bn ∞ diverges,
then
 ∑a n =1 ∞ ∑a n =1 ∑a ∞ ∑a n =1 ∞ ∑b ∞ 5.
If
 n 
diverges
(k
≠
0)
 n =1 ∞ ∑a n =1 ∑ ka n 
both
diverge,
it’s
inconclusive.
 n 
 ± bn 
 
 Do:

Determine
if
each
of
the
following
series
converge
or
 diverge.
 ∞ 1.
 ∑ n =1 ∞ en 
 31 +n 2.
 ∑ n 3
 n =1 ∞ 3 3 − 3.
 ∑ n n − 3
 n =4 ...
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