Sum of an infinite series

In summary, the conversation discusses the task of proving that the sum of an infinite series is irrational. The participants consider the properties of rational numbers and explore the idea of representing the number in different bases. They also mention the use of Liouville's Approximation Theorem and Dirichlet's Theorem to construct proofs for the irrationality and transcendence of certain numbers. However, they conclude that the original problem may not be able to be solved using elementary methods like Euclid's proof for the irrationality of the square root of 2.
  • #1
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i have to prove that the sum of the follwing infinite series is irrational.

[tex]\frac{1}{2^3} + \frac{1}{2^9} + \frac{1}{2^{27}} + \frac{1}{2^{81}} + \cdots[/tex]

i have no idea where to begin. thanks in advance.

note: this is not a homewrok problem.
 
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  • #2
What's true about the decimal expansion of rational numbers?
 
  • #3
the decimal expansion of a rational number either terminates or repeats a pattern. but how does that help?
 
  • #4
murshid_islam said:
the decimal expansion of a rational number either terminates or repeats a pattern. but how does that help?
Is that fact unique to a decimal expansion?
 
  • #5
murshid_islam said:
the decimal expansion of a rational number either terminates or repeats a pattern. but how does that help?

That doesn't necessarily help you FIND the sum, but your problem is to prove the number is irrational. What happens if you think of this as a number written in binary?
 
  • #6
HallsofIvy said:
That doesn't necessarily help you FIND the sum, but your problem is to prove the number is irrational. What happens if you think of this as a number written in binary?
it becomes 0.001000001000000000000000001...
clearly it doesn't terminate and no pattern repeats. but does this fact of irrational numbers hold true for any base?
 
  • #7
Of course it does. If a number, in a rational base, has a terminating decimal expansion, then that is true of any other rational base. Obviously you can get a terminating decimal for pi in base pi...
 
  • #8
Fine recipe for proving that real numbers represented by sums of murshid's kind are irrational or even trancendent is http://mathworld.wolfram.com/LiouvillesApproximationTheorem.html" [Broken]

:smile:
 
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  • #9
The "general" sketch for proving irrionality is to express the sum as S = a/b

and then prove that if that was the case, then

x<a<x+1

where x is some integer and therefore there is contradiction; an integer cannot lie between two integers.
 
  • #10
Werg22 said:
The "general" sketch for proving irrionality is to express the sum as S = a/b

and then prove that if that was the case, then

x<a<x+1

where x is some integer and therefore there is contradiction; an integer cannot lie between two integers.

The most common form I see is expressing it as a/b in lowest terms and then showing that a/b = c/d with 0 < c < a.
 
  • #11
Werg22 said:
The "general" sketch for proving irrionality is to express the sum as S = a/b

and then prove that if that was the case, then

x<a<x+1

where x is some integer and therefore there is contradiction; an integer cannot lie between two integers.

CRGreathouse said:
The most common form I see is expressing it as a/b in lowest terms and then showing that a/b = c/d with 0 < c < a.

But, in this particular case that would be much more difficult that just observing that the number, in base 2, is neither a terminating nor a repeating "decimal".
 
  • #12
Werg22,CRGreathouse:

Of course,"Reductio ad absurdum" is ,pretty often, a logical way of organization of the proofs.Unfortunately,as HallsofIvy indicates, it doesn't tell anything on how to construct proofs from case to case!
It may appear you two sound like suggesting there is elementar proof in this case,something like [itex]\sqrt{2}[/itex] is irrational?
Please,show your proofs ,I'm not aware this can be done at elementar level.
Proofs that real numbers :
[tex]\sum_{n=1}^{\infty}\frac{1}{2^{(n!)}}[/tex]
Or
[tex]\sum_{n=1}^{\infty}\frac{1}{2^{(m^n)}};m\geq 2[/tex]

are irrational and transcendent can be constructed by employment of joined consequences of Dirichlet's Theorem and mentined above Liouville's Theorem.I don't think I can classify Lioville's Tm by elementar label...
 
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  • #13
What do you mean by "elementary proof"?

Euclid's proof seems pretty elementary to me: If [itex]\sqrt{2}[/itex] is rational then [itex]\sqrt{2}= \frac{m}{n}[/itex], reduced to lowest terms. Then [itex]2= \frac{m^2}{n^2}[/itex] so [itex]2n^2= m^2[/itex]. But the square of any odd number is odd ((2n+1)2= 4n2+ 4n+ 1= 2(2n2+ 2n)+ 1). Since m2 is even, m must be even: m= 2k for some integer k. Then [itex]2n^2= (2k)^2= 4k^2[/itex] and so [itex]n^2= 2k^2[/itex]- that is, n is also even, contradicting the fact that m and n are relatively prime.

A direct proof, somewhat less "elementary" is this: [itex]\sqrt{2}[/itex] obviously satisfies the equation x2- 2= 0. By the "rational root theorem", any rational root of that equation must have numerator that evenly divides the constant term, 2, and denominator that evenly divides the leading coefficient 1. The only possible rational roots, then, are 2 and -2, neither of which satisfies the equation. x2- 2= 0 has no rational roots so [itex]\sqrt{2}[/itex].

That, of course, has little to do with the original question, but I like to show off!:approve:
 
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  • #14
HallsofIvy said:
What do you mean by "elementary proof"?
The one that doesn't rely on theorems from math analysis (complex analysis in particular.)
Read my post more carefully.I'm not questioning if the irrationality of [itex]\sqrt{2}[/itex] can be prooven by elementar math methods,but if OP's problem can be solved in similar way.
I think it can't (but you can never know for sure :smile:).
If you can solve problems ,and prove irrationality (let alone transcendence) of the numbers represented by the sums given above,in "Euclid's" fashion ,post your solution.We would like to know.
 

What is the sum of an infinite series?

An infinite series is a sum of an infinite number of terms. The sum of an infinite series is the sum of all the terms in the series, which can be finite, infinite, or undefined.

How do you calculate the sum of an infinite series?

The sum of an infinite series can be calculated using various methods such as the geometric series formula, telescoping series, or the ratio test. It is important to check for convergence before attempting to calculate the sum.

What is the difference between an infinite series and a finite series?

A finite series has a fixed number of terms, while an infinite series has an unlimited number of terms. This means that the sum of a finite series is a finite number, while the sum of an infinite series can be a finite, infinite, or undefined number.

What is the importance of understanding infinite series in mathematics?

Infinite series are used in various mathematical fields, such as calculus, differential equations, and number theory. They also have practical applications in physics, engineering, and economics. Understanding infinite series is essential for solving complex problems and understanding real-world phenomena.

Are there any real-life examples of infinite series?

Yes, there are many real-life examples that can be modeled using infinite series, such as compound interest, population growth, and radioactive decay. These examples show the practical applications of infinite series in various fields.

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