Infinite Geometric series?

Question

how to prove the formula :

a
_______
1 - r


And why are there error involved like using this one can prove that 0.9................ ===== 1.0......

Answer 1

First, it's true that 0.999999999999999999.... = 1.

If you don't believe that, then calculate the value of
1 - 0.999999999999999...
It's 0.00000000000...
So there is no difference between 1 and 0.999999999999...

Now, the equation you ask about is:
a + ar + ar^2 + ar^3 + ... = a / (1 - r)

The proof is as follows.
Let S (for "sum") = a + ar + ar^2 + ar^3 + ...
Then rS = ar + ar^2 + ar^3 + ...
And S - rS = a (because all the other terms cancel).

Factoring, S (1 - r) = a
Dividing, S = a / (1 - r)

Answer 2

a1 = 0.9
a2 = 0.09 which is a1 times 0.1
a3 = 0.009 which is a2 times 0.1

so r = 0.1 and a1 = 0.9

So the sum is 0.9/(1 - 0.1) = 0.9/0.9 = 1 which is the right answer

because 0.9 repeating does equal 1.