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)