Let's consider the alternating sign series...

(1)

If we indicate with the general term of (1), you can verify that...

1) is a decreasing sequence

2)

... so that (1) converges. Now, indicating with the sum of (1), the questions I intend to indicate to your attention are...

a) is rational or irrational?...

b) if is irrational, can it be expressed as elementary function of rational numbers or other constants like , and so one?...

Kind regards