?The meaning of the decimal representation of a number \(0 . d_{1} d_{2} d_{3} \ldots\)
Chapter 10, Problem 45(choose chapter or problem)
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
\(0 \cdot d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\ldots\)
Show that this series converges for all choices of \(d_{1}, d_{2}, \ldots\)
Equation Transcription:
0, 1,
Text Transcription:
0.d_1d_2d_3…
d_1
0, 1, 2, . . . , 9)
0.d_1d_2d_3d_4 . . . = d_1/10 + d_2/10^2 + d_3/10^3 + d_4/10^4 + . . .
d_1,d_2, . . . .
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