Determine if each infinite geometric series is convergent or divergent?

First figure out the common ratio by taking any term and dividing it by the prior term.

If that ratio (r) is less than 1, then the series converges to a finite sum. (Technically you want to say |r| < 1 but all your ratios are positive.)

Let's check a few:
0.03 / 0.3 = 0.1 < 1 ← converges
0.05 / 0.25 = 0.2 < 1 ← converges
1.15 / 0.23 = 5 ≥ 1 ← diverges
...

Incidentally, if |r| < 1, then the formula for the infinite sum is:
S = a/(1 - r)

a : first term
r : common ratio

For example, the first series converges to:
S = 0.3 / (1 - 0.1)
S = 0.3 / 0.9
S = 3/9
S = 1/3

And it makes sense if you think of it as a decimal:
S = 0.3 + 0.03 + 0.003 + ...
S = 0.33333... 
S = 1/3

I believe that if the common ratio is:

if 0 < |r| < 1 then it's divergent
if |r| > 1 then it's convergent

So divide the second term by the first to get the common ratio:

0.03 / 0.3 = 0.1 (divergent)

0.05 / 0.25 = 0.2 (divergent)

1.15 / 0.23 = 5 (convergent)

etc.