The meaning of the decimal representation of a number 0.d1d2d3 . . (where the digit i is one of the numbers 0, 1, 2, . ., 9) is that 0.d1d2d3d4 . . = d1/10 + d2/10^2 + d3/10^3 + d4/10^4 + . . 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