(November 26, 2022 at 5:54 pm)LinuxGal Wrote:(November 26, 2022 at 5:17 pm)polymath257 Wrote: Step 1: Prove the expression .999.... makes sense.
Otherwise, you could argue as follows:
x=1+2+4+8+16+...
2x=2+4+8+16+32+...
Hence,
x=1+2x
so
-x=1
x=-1
In particular,
1+2+4+8+... <0.
Not so, since x = 0.999... has an upper bound as the decimal expansion continues without limit, but x = 1+2+4... diverges.
And *that* is the point. A discussion about convergence and what, precisely, a decimal expansion means in terms of limits.
Once the definition is given, the equality is automatic. No algebraic manipulations required. Just the fact that 1/10^n --> 0 as n-->infty.