Answer :
This series 9 (.d1d2d3 . . .) is convergent because it is in increasing order.
How to describe the convergent series?
- In a geometric progression, each element following the first is created by multiplying it by a number known as the common ratio, which is represented by the symbol r.
For the stated question-
- The information provided indicates that the decimal expression's digits, d, are 10.
- The common ratio is 1/10, and the right-hand side of a inequality is a sum with 1 as the first term in the geometric progression.
The geometric progression's sum in this instance is represented as follows:
= a / (1 - r)
= 1 / (1 - r)
= 10/9
As a result, the series becomes convergent since it is in increasing order.
To know more about the convergent series, here
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The complete question is-
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.
