Real Analysis, Sequences in relation to Geometric Series and their sums

In summary, the sequence of integers with for each n such that is unique except when x is of the form , in which case there are exactly two such sequences.
  • #1
joypav
151
0
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.
 
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  • #2
For a start, let be the greatest integer such that . Having chosen , let be the greatest integer such that . Show that the sequence of partial sums converges to .
 
  • #3
Euge said:
For a start, let be the greatest integer such that . Having chosen , let be the greatest integer such that . Show that the sequence of partial sums converges to .

Then we have,



Take the ,


converges to x.

Is this correct? Now to show uniqueness do I proceed by way of contradiction?
 
  • #4
Also, I don't understand the second part of the problem either. Now I am to find the x it converges to?
 

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