The meaning of the decimal representation of a number 0.d1d2d3 (where the digit di is one of the numbers 0, 1, 2, , 9) is that 0.d1d2d3d4 = d1 10 + d2 102 + d3 103 + d4 104 + show that this series always converges.

Ratio testwe can show it converges absolutely lim          |d{n+1}/10^(n+1)|/|d{n}/10^n|n->inf=  "          |d{n+1}/10^n*10|*|10^n/d{n}|=  "          1/10*|d{n+1}/d{n}|the digits can range form 0 to 9the biggest number that can be in that absolute value thing is 9/1so1/10*9=9/10<1so it converges absolutely