geometric series

Geometric Series
The series
?_(k=0)^???ar?^k =a+ar+?ar?^2+?ar?^3+?+?ar?^k+?
is called a geometric series with common ratio r.
Let us form the sequence of partial sums:
S_0=a
S_1=a+ar
S_2=a+ar+?ar?^2
S_3=a+ar+?ar?^2+?ar?^3
and so forth. In general
S_n=a+ar+?ar?^2+ar^3+?+?ar?^n
Multiplying S_n by r gives
?rS?_n=ar+ar^2+?ar?^3+?+?ar?^(n+1)
S_n-?rS?_n=a-?ar?^(n+1)
(1-r)S_n=a(1-r^(n+1) )
S_n=a(1-r^(n+1) )/(1-r) ,r?1
Cases of r :
1.If |r|<1, then

?lim??(n??)??r^(n+1) ?=0 ,

and
?lim??(n??)??S_n ?=a/(1-r),
And the series converges.
2. If |r|>1, then
?lim??(n??)??r^(n+1) ?=? ,
and
?lim??(n??)??S_n ?=?
And the series diverges.

If r=-1, then

S_n=(1-(-1)^(n+1))/2={?(0,n odd@1,n even)?
And the series diverges.

If r=1 , the series becomes
1+1+1+?+1+?=?_(k=1)^??1
S_n=1+1+?+1=n+1
and
?lim??(n??)??S_n ?=?
And the series diverges.
Geometric Series
The series
?_(k=0)^???ar?^k =a+ar+?ar?^2+?ar?^3+?+?ar?^k+?
is called a geometric series with common ratio r.
Let us form the sequence of partial sums:
S_0=a
S_1=a+ar
S_2=a+ar+?ar?^2
S_3=a+ar+?ar?^2+?ar?^3
and so forth. In general
S_n=a+ar+?ar?^2+ar^3+?+?ar?^n
Multiplying S_n by r gives
?rS?_n=ar+ar^2+?ar?^3+?+?ar?^(n+1)
S_n-?rS?_n=a-?ar?^(n+1)
(1-r)S_n=a(1-r^(n+1) )
S_n=a(1-r^(n+1) )/(1-r) ,r?1
Cases of r :
1.If |r|<1, then

?lim??(n??)??r^(n+1) ?=0 ,
and
?lim??(n??)??S_n ?=a/(1-r),
And the series converges.
2. If |r|>1, then
?lim??(n??)??r^(n+1) ?=? ,
and
?lim??(n??)??S_n ?=?
And the series diverges.

If r=-1, then

S_n=(1-(-1)^(n+1))/2={?(0,n odd@1,n even)?
And the series diverges.

If r=1 , the series becomes
1+1+1+?+1+?=?_(k=1)^??1
S_n=1+1+?+1=n+1
and
?lim??(n??)??S_n ?=?
And the series diverges.