Question

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.

Answer 1

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 

Answer 2

The series is in increasing order, therefore the series is convergent.

How to explain the series.

In the information given, d < 10, being the digits in decimal expression.

The right hand side of the inequality is a sum with the first term of the geometric progression being 1 e the common ratio is 1/10.

In this case, the sum of the geometric progression is given as:

= a / (1 - r)

= 1 / (1 - r)

= 10/9

Therefore, the series is in increasing order, therefore the series is convergent.

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