3.1 Sequences
definition:
a sequence is an infinite listindexed by IN 51,2, 3....3 Covany [a,a +1, a +2. I often
=
...
starting at0
Notation:
C., C2, C3...
ECn3,.. or [Cn]
eg. Cn in"
for nyl explicitformula
=
Emma
(n nCn-1, for ne formula
=
recursive
(, 1, (n
=
2.1,
=
2z 3)2
=
3.2.1
=
(n n(n-1)....
=
3.2
eg. An=(-1)" or An=Cos(in) for n =0
1, -1,1,-.
a sequence [an3 converges to limit ((himAn=1) ifwe can make an as close as we can
to L by requiring to be sufficiently large.
dany diverges otherwise ( An dne)
eg. 0Un=cos (in)
ing (OS (in) dne
② lz 0
=
Sins converges to 0
if f is defined on R (or any continuous domain), **f(x) 2
=
and if an=f(n), then
liAn L
=
eg. O Re
x
0
-
=
f(x) e
=
② arctan(n) Fz =
③ In (In) = -
-(dne)
Sen(/n) 3 diverges
all limit rules apply provided all limits existand the limitofthe denominator of a quotient to
ifan=br? In for all n and han=(n=2 then limba c
=
ifg is continuous and Ran=2 then ig(an) g() =
eg. 02 Arctan (In)=arctan (0) 0
=
② An=
I
-
In an in en =0, 3 converges 0
to
!
~Ov
definition:
a sequence is an infinite listindexed by IN 51,2, 3....3 Covany [a,a +1, a +2. I often
=
...
starting at0
Notation:
C., C2, C3...
ECn3,.. or [Cn]
eg. Cn in"
for nyl explicitformula
=
Emma
(n nCn-1, for ne formula
=
recursive
(, 1, (n
=
2.1,
=
2z 3)2
=
3.2.1
=
(n n(n-1)....
=
3.2
eg. An=(-1)" or An=Cos(in) for n =0
1, -1,1,-.
a sequence [an3 converges to limit ((himAn=1) ifwe can make an as close as we can
to L by requiring to be sufficiently large.
dany diverges otherwise ( An dne)
eg. 0Un=cos (in)
ing (OS (in) dne
② lz 0
=
Sins converges to 0
if f is defined on R (or any continuous domain), **f(x) 2
=
and if an=f(n), then
liAn L
=
eg. O Re
x
0
-
=
f(x) e
=
② arctan(n) Fz =
③ In (In) = -
-(dne)
Sen(/n) 3 diverges
all limit rules apply provided all limits existand the limitofthe denominator of a quotient to
ifan=br? In for all n and han=(n=2 then limba c
=
ifg is continuous and Ran=2 then ig(an) g() =
eg. 02 Arctan (In)=arctan (0) 0
=
② An=
I
-
In an in en =0, 3 converges 0
to
!
~Ov
