Find the sum of the infinite series 1/3+4/9+16/27+64/81+ if it exists.

9/27, 12/27, 16/27

So this is a geometric sequence as each term is 4/3 the previous term.

Since the common ration is greater than one the sum of the series diverges, it does not exist.  (The sum just keeps getting larger and larger)

For a geometric series to have a sum r^2<1

So that the normal sum

s(n)=a(1-r^n)/(1-r)   becomes if r^2<1

s=a/(1-r)