Given the string dn=1+2+...2^n evaluate limit of dn/2^n if n->+infinite

We notice that the terms of the sum are the terms of a
geometric progression, whose common ratio is r = 2.

The sum
of n terms of a geometric progression is:

dn = 1*(2^n -
1)/(2 - 1)

dn = (2^n - 1)

Now,
we can evaluate the limit;

lim dn/2^n = lim (2^n -
1)/2^n

We'll get:

lim
(2^n)/2^n - lim 1/2^n

We'll simplify and we'll
get:

lim 1 - lim 1/2^n

since
n->+infinite => lim 2^n = infinite => lim 1/2^n =
0

lim dn/2^n = 1 -
0

lim dn/2^n = 1, for n
-> +infinite