- #1
joypav
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I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it.
Problem:
Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence of integers with for each n such that
and that this sequence is unique except when x is of the form , in which case there are exactly two such sequences.
Show that, conversely, if is any sequence of integers with , the series
converges to a real number x with . If p=10, this sequence is called the decimal expansion of x. For p=2 it is called the binary expansion of x, and for p=3, the ternary expansion.
Problem:
Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence
and that this sequence is unique except when x is of the form
Show that, conversely, if
converges to a real number x with