Question

How to prove the formal to find the infinite geometric series is correct?

Answer 1

note that
(1-r^n) = (1-r)(1 + r + r^2 + r^3 + .....)
as n→∞ 1-r^n → 1
and the second factor on the right is the sum of the infinite series.

see what you can do with that.

Answer 2

I still don't understand :(

Answer 3

c'mon -- the sum is
a + ar + ar^2 + ar^3 + ...
= a (1 + r + r^2 + r^3 + ...)
now recall that
(1-r^n) = (1-r)(1 + r + r^2 + r^3 + ...)
so the sum is the limit as n→∞ of
a(1-r^n)/(1-r) = a/(1-r)

if you're still unconvinced, you can find more proofs online.
Or probably in your class text.