The numbers in the denominators are exactly those numbers that have either 2 or 5 or both as their prime factors, eg 8 = 2x2x2, 10=2x5. This is exactly the property which means they have a terminating decimal expansion.

You can organise the factions into a different order such that you first add those with no 5 factor at all:

A = (1 + 1/2 + 1/4 + 1/8 + 1/16 + …)

If you first double it:

2A = (2 + 1 + 1/2 + 1/4 + 1/8 + 1/16 + …)

And then you subtract the original:

A = 2

Next you look at those fractions which have exactly 1 factor of 5:

(1/5 + 1/10 + 1/20 + 1/40 + 1/80 + …)

You can factorise the 1/5 out of each term:

1/5 x (1 + 1/2 + 1/4 + 1/8 + 1/16 + …)

But that is just: 1/5 x A

The next group will be fractions with exactly 2 factors of 5, and similarly will total to 1/25 x A

The entire sequence will total:

AB, where B = (1 + 1/5 + 1/25 + 1/125 + …)

Multiply B by 5:

5B = (5 + 1 + 1/5 + 1/25 + 1/125 + …)

And subtract B:

4B = 5, so B = 5/4

The overall sum AB is therefore 2 x 5/4 = 5/2 = 2.5