3.2 Intro to Series
Definition:
series
A a sum of infinitely many terms;(a, + az azt...)
+
notation: ne, An
eg. 0neikn( k 4
=
+
6 4x+ ...)
+ +
② 0.333...
...
-
=
Bon
=
③ Definite integral! (...)
n
·
- (...)
k1 =
Given a series, mean, we define the nth partial sum of the series to be Sn aj
=
A,
=
taztAgt...
ifthe
sequence [Sn3 converges and **Sn=5 then we say the series,an converges to 5 and we
writen =,An S =
if ESn3 diverges, no,an diverges
eg. 0An= (-1)", e,)-1) = ?
Sn 2,) 1) 1 1 1 1...1)
S 0,
-
his even
-
= = + - +
=
-
, n is odd
↳MoSn dne, ESn] diverges
Ian ?,(-1)" =
diverges
② Suppose the
&
nth partial sum ofa series,an is
Sn=5-an
a) whatis n=,An? divide each termby n
hi Sn (5
=
-
) (S-3in
-
=
5-43 is
=
so ,an 1/3
=
(convergent)
b) Whatis an?
Sn a, Act
= +
. . . An-itAn
Sn-(Sn-1) An =
(
an=
($-3ny)-15-an-is
1) 3(n -
4
+
2(n 1)
-
-
3(n 1) 4
+
-
③Yo (1+
=
r +
r2 r3+ +
...)
Fr 1
=
-r* for rel
1- r
1
Sn 2 =un+1-1
+
1
for re
-
=
1 -
r r -
d
, inr
E ,if Ir)< otherwise
1
0 if and
diverges
=
r 1
=
&
eifum
e
1
fn
+
1
Iris
8
-
if
1 - r
ifIr)< 1 and
r diverges otherwise
-
nEor= I
if (t) <
1
"geometric series"
1 -
r
geometric sum
-
eg. itn n=o(t)" r tz
=
=
-I 2
=
1 -
t
·
In?
1
n =
Eit otr-Stermwhere =
n 0)
=
t
2 1 1
-
- -
- = =
② n=Yer no(ter) =
geometric series with Ir)= be:
e
- te
-
=
-2 -
1
③_E52 n=0(-43) =
-
8/3 1- 83) 8/31 =
Is
=
divergea
Another case,where you find
a formula for Sn:"telescoping sums"
eg. E, n(n+1)
Note
mentistint
Sn x=i)k
=
-
x 1) +
-
- et
substitute k+1 for t
~l
22
-Eite-e
=
e=2 to
Sn = 1 (for everything cancels) -
I
n+1
( + 1) 1
in =
(1
-
=
&
n ,n(n 1)
=
+
eg.n ·fi)n3
0
=
+3
-
=ni(ts)" 3(55) +
Definition:
series
A a sum of infinitely many terms;(a, + az azt...)
+
notation: ne, An
eg. 0neikn( k 4
=
+
6 4x+ ...)
+ +
② 0.333...
...
-
=
Bon
=
③ Definite integral! (...)
n
·
- (...)
k1 =
Given a series, mean, we define the nth partial sum of the series to be Sn aj
=
A,
=
taztAgt...
ifthe
sequence [Sn3 converges and **Sn=5 then we say the series,an converges to 5 and we
writen =,An S =
if ESn3 diverges, no,an diverges
eg. 0An= (-1)", e,)-1) = ?
Sn 2,) 1) 1 1 1 1...1)
S 0,
-
his even
-
= = + - +
=
-
, n is odd
↳MoSn dne, ESn] diverges
Ian ?,(-1)" =
diverges
② Suppose the
&
nth partial sum ofa series,an is
Sn=5-an
a) whatis n=,An? divide each termby n
hi Sn (5
=
-
) (S-3in
-
=
5-43 is
=
so ,an 1/3
=
(convergent)
b) Whatis an?
Sn a, Act
= +
. . . An-itAn
Sn-(Sn-1) An =
(
an=
($-3ny)-15-an-is
1) 3(n -
4
+
2(n 1)
-
-
3(n 1) 4
+
-
③Yo (1+
=
r +
r2 r3+ +
...)
Fr 1
=
-r* for rel
1- r
1
Sn 2 =un+1-1
+
1
for re
-
=
1 -
r r -
d
, inr
E ,if Ir)< otherwise
1
0 if and
diverges
=
r 1
=
&
eifum
e
1
fn
+
1
Iris
8
-
if
1 - r
ifIr)< 1 and
r diverges otherwise
-
nEor= I
if (t) <
1
"geometric series"
1 -
r
geometric sum
-
eg. itn n=o(t)" r tz
=
=
-I 2
=
1 -
t
·
In?
1
n =
Eit otr-Stermwhere =
n 0)
=
t
2 1 1
-
- -
- = =
② n=Yer no(ter) =
geometric series with Ir)= be:
e
- te
-
=
-2 -
1
③_E52 n=0(-43) =
-
8/3 1- 83) 8/31 =
Is
=
divergea
Another case,where you find
a formula for Sn:"telescoping sums"
eg. E, n(n+1)
Note
mentistint
Sn x=i)k
=
-
x 1) +
-
- et
substitute k+1 for t
~l
22
-Eite-e
=
e=2 to
Sn = 1 (for everything cancels) -
I
n+1
( + 1) 1
in =
(1
-
=
&
n ,n(n 1)
=
+
eg.n ·fi)n3
0
=
+3
-
=ni(ts)" 3(55) +
