The meaning of the decimal representation of a number 0 . d1...

The meaning of the decimal representation of a number $0 . d_{1} d_{2} d_{3} \ldots$. (where the digit $d_{i}$ is one of the numbers 0,1 , $2, \ldots, 9$ ) is that $$0 . d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\cdots$$ Show that this series converges for all choices of $d_{1}, d_{2}, \ldots .$

The meaning of the decimal representation of a number $0 . d_{1} d_{2} d_{3} \ldots$. (where the digit $d_{i}$ is one of the numbers 0,1 , $2, \ldots, 9$ ) is that
$$0 . d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\cdots$$
Show that this series converges for all choices of $d_{1}, d_{2}, \ldots .$

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Key Concepts

Geometric Series and Its Convergence

A geometric series is a series in which each term is a constant multiple (the common ratio) of the previous term. A key result is that a geometric series converges if the absolute value of the common ratio is less than one. This principle can be used by comparing another series to a geometric series to establish its convergence.

Comparison Test

The comparison test is a method for establishing the convergence of an infinite series by comparing its terms to those of a second series whose convergence properties are already known. If the terms of the original series are bounded by the corresponding terms of a convergent series, then the original series also converges.

Infinite Series Convergence

This concept involves determining whether the sum of an infinite sequence of terms converges to a finite limit. In the context of real analysis, various tests and comparisons can be applied to decide if a series converges or diverges, and a series converges if its sequence of partial sums has a finite limit.

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