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

• MHB
• joypav
In summary, the sequence $(a_n)$ of integers with $0 \leq a_n < p$ for each n such that $x = \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n}$ is unique except when x is of the form $\frac{q}{p^n}$, in which case there are exactly two such sequences.
joypav
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 $(a_n)$ of integers with $0 \leq a_n < p$ for each n such that
$x = \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n}$

and that this sequence is unique except when x is of the form $\frac{q}{p^n}$, in which case there are exactly two such sequences.

Show that, conversely, if $(a_n)$ is any sequence of integers with $0 \leq a_n < p$, the series

$\sum\limits_{n=1}^{\infty} \frac{a_n}{p^n}$

converges to a real number x with $0 \leq x \leq 1$. 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.

For a start, let $a_1$ be the greatest integer such that $\dfrac{a_1}{p} \le x$. Having chosen $a_1,\ldots, a_{N-1}$, let $a_N$ be the greatest integer such that $\displaystyle \sum_{n = 1}^N \frac{a_n}{p^n} \le x$. Show that the sequence of partial sums $\displaystyle \left(\sum_{n = 1}^N \frac{a_n}{p^n}\right)_{N=1}^\infty$ converges to $x$.

Euge said:
For a start, let $a_1$ be the greatest integer such that $\dfrac{a_1}{p} \le x$. Having chosen $a_1,\ldots, a_{N-1}$, let $a_N$ be the greatest integer such that $\displaystyle \sum_{n = 1}^N \frac{a_n}{p^n} \le x$. Show that the sequence of partial sums $\displaystyle \left(\sum_{n = 1}^N \frac{a_n}{p^n}\right)_{N=1}^\infty$ converges to $x$.

Then we have,

$0 < x - \sum\limits_{n=1}^{N} \frac{a_n}{p^n} < \frac{1}{p^N}$

Take the $lim_{n\rightarrow{\infty}}$,
$\vert x - \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n} \vert = 0$

$\implies \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n}$ converges to x.

Is this correct? Now to show uniqueness do I proceed by way of contradiction?

Also, I don't understand the second part of the problem either. Now I am to find the x it converges to?

1. What is real analysis?

Real analysis is a branch of mathematics that deals with the study of real numbers, their properties, and the functions defined on them. It involves the use of concepts such as limits, continuity, differentiation, and integration to analyze and understand the behavior of real-valued functions.

2. What are sequences in relation to geometric series?

A sequence is a list of numbers that follow a specific pattern or rule. In geometric series, each term is obtained by multiplying the previous term by a fixed constant called the common ratio. This creates a sequence of numbers that follow a geometric pattern.

3. How are sequences and geometric series related to sums?

In real analysis, the sum of an infinite geometric series can be calculated using a formula that involves the first term, the common ratio, and the limit of the series as the number of terms approaches infinity. This allows us to determine the sum of a sequence that follows a geometric pattern.

4. What is the significance of geometric series and their sums in real analysis?

Geometric series and their sums are essential in real analysis as they provide a way to approximate functions and solve problems related to limits and convergence. They also have applications in various fields, such as physics, engineering, and finance.

5. How can geometric series and their sums be used to solve real-world problems?

Geometric series and their sums have various applications in the real world, such as calculating compound interest, determining the growth rate of populations, and analyzing the behavior of electrical circuits. They can also be used to approximate functions and solve differential equations, making them valuable tools in many scientific and engineering fields.

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