What do you think of this infinite series?

Its a cool, nice series. But do you have a math question related to it?

I'm assuming the 'x' was to represent multiplication? I think you are trying to write the following infinite geometric sum.
∞
∑ (0.9) * (0.1)^n
n=0

= 0.9 + 0.09 + 0.009 + 0.0009 + ...
= 0.9999...

That's a geometric series with 0.9 as the first term and 0.1 as the common ratio.

The formula for the sum of an infinite series is:
S = a/(1 - r), as long as |r| < 1

a : 0.9
r : 0.1

S = 0.9 / (1 - 0.1)
S = 0.9 / 0.9
S = 1

Answer:
0.999... = 1

Let's rewrite it as

(9/10)x*(1 + 1/10 + 1/100 + . . .)

=(9/10)x * (10/9) = x

(0.9x)(0.1)^n, n = 0 to infinity
sum_(n=0)^∞ 0.9 0.1^n = 1