Find the sum of the infinite series

1/3

+

4/9

+

16/27

64/81 ... if it exists.

It does not exist.

Step-by-step explanation:

If we divide the second term by the first, 3rd by the 2nd and the 4th by the 3rd we get a constant result 4/3.

So this is a Geometric Series with common ratio r = 4/3 and a1 = 1/3.

As the common ratio is greater than 1 the series diverges (continues to get bigger) so there is no sum to infinity. It just goes on getting bigger and bigger without bounds.