Number between less than 15 or equal to 15 or between the two options?

Question

Is $14.9999999$ ad infinitum $\lt 15$, $= 15$, or in between?

Answer 1

It is exactly $15$, in the same way that $0.999 \ldots = 1$. For my second affirmation, notice that $1/3 = 0.333\ldots$, then $1 = 3/3 = 3 \cdot 0.333\ldots = 0.999\ldots$.

Answer 2

Definitely equal. It's a sum of a geometric series:

$$14.9999\cdots =14+\frac9{10}\left(1 + \frac1{10}+\frac1{10^2}+\cdots\right) $$

$$=14+\frac9{10}\left(\frac1{1-\frac1{10}}\right) $$

$$=14+\frac9{10}\left(\frac{10}9\right) $$

$$=14+1$$

$$=15$$

as desired.