Proving 10+100+1000+10000+100000+...=-(1/9) - Any Ideas?

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In summary, the conversation discusses a mathematical statement that claims the sum of an infinite series, 10+100+1000+10000+100000+..., is equal to a negative number, -(1/9). The group discusses different "proofs" for this statement, including using geometric progression and subtracting infinite numbers. However, they ultimately conclude that this statement is not mathematically true due to the series not converging and the impossibility of a sum of positive numbers equalling a negative number. They also mention a professor giving a talk about a similar statement in 2-adic numbers.
  • #1
sutupidmath
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we have to prove that

10+100+1000+10000+100000+...=-(1/9)

any ideas?
 
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  • #2
maybe you are trying to "prove" that
10+100+1000+10000+100000+...=-(10/9)

here is a "proof" :)

let
S = 10 + 100 + 1000 + 10000 + ...

then
10S = 100 + 1000 + 10000 + ...

now,
S - 10S = 10
=> -9S = 10
=> S = -(10/9)
=> 10 + 100 + 1000 + 10000 + ... = -(10/9)
 
  • #3
i don't know maybe this is what i actually was looking for.
 
  • #4
but note that what i have given as a "proof" is not really a proof at all. the series 10 + 100 + 1000 + ... doesn't converge. so my "proof" doesn't actually work.
 
  • #5
murshid_islam said:
but note that what i have given as a "proof" is not really a proof at all. the series 10 + 100 + 1000 + ... doesn't converge. so my "proof" doesn't actually work.

so Murshid_islam what is the deal here? I can see that the series does not converge, however where is the problem on your proof? Is there a mathematical error, cause i could not see it, or what can we say about this?
 
  • #6
The error was when he subtracted the two and got a fixed real number. Subtracting infinity from infinity is not a well-defined operation.
 
  • #7
Because, in his expression for 10S, he ignored the largest term present there.
 
  • #8
as long as we are talking for infinit large numbers i cannot grasp how could there be a larger number on 10S than on S.I think it is absurd to talk about a "largest"term here, as long as we deal with infinit large terms! however i do understand the error now. SO defenitely we can say that
10+100+1000+10000+100000+...=-(10/9)

is not mathematically true, and i cannot count on it, right?
 
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  • #9
I would have thought it obvious from the start that a sum of positive numbers cannot be negative!

Yes, I recommend that you not count on it!
 
  • #10
why don't you use sum of infinite G.P?
 
  • #11
What does G.P mean at first place? I am sorry i am not used to these, so i really don't know what they stand for?
can you tell me?
 
  • #12
HallsofIvy said:
I would have thought it obvious from the start that a sum of positive numbers cannot be negative!

!

Yeah, i also thought it could not be negative. However i saw this on a tv scientific show, and a proffesor demonstrated this, so i just wondered how that would be possible. That proffesor, whose name i cannot remember, said that he had turned this for a mathematical test to prove that this is right. If ,at first place, this is exactly what i saw, couse i am not 100% posotive.
 
  • #13
sutupidmath said:
What does G.P mean at first place? I am sorry i am not used to these, so i really don't know what they stand for?
can you tell me?

G.P - Geometric Progression

It is a series in which each term, apart from the first, is a fixed multiple of the previous term.

a + ar + ar^2 + ar^3 + ...+ar^n+...

The sum of the first n terms of such a series is a(1-rn)/(1-r). Check what happens for your series, when n tends to infinity.
 
  • #14
thnx, i do know what a geometric progression is, but just did not know that g.p stands for that.
thnx indeed.
 
  • #15
i think after we find the sum of that geometric progression using a(1-rn)/(1-r), and if we evaluate the limit of the result, it turns out that the sum must be infinity. Is that right?
 
  • #16
sutupidmath said:
i think after we find the sum of that geometric progression using a(1-rn)/(1-r), and if we evaluate the limit of the result, it turns out that the sum must be infinity. Is that right?
Yes, it diverges.

But, as mentioned earlier, the thing that should first convince you that the statement is not true is that the a sum of positive numbers cannot give you a negative number.
 
  • #17
yeah, thank you guys for your help.
 
  • #18
sutupidmath said:
so Murshid_islam what is the deal here? I can see that the series does not converge, however where is the problem on your proof? Is there a mathematical error, cause i could not see it, or what can we say about this?
the error was when i let S = 10 + 100 + 1000 + ...
As the series doesn't converge i cannot let it equal to a number S.
 
  • #19
Funny coincidence, but this Friday a prof at m university is giving a talk on why 1+2+4+6+...=-1 in the 2-adic numbers.
 

FAQ: Proving 10+100+1000+10000+100000+...=-(1/9) - Any Ideas?

How can you prove that 10+100+1000+10000+100000+...=-(1/9)?

The proof for this equation involves using a mathematical concept called an infinite geometric series. This series involves a constant ratio (r) that is multiplied by the previous term to get the next term. In this case, the ratio is 10, and the first term is 10. So, the sum of the series can be represented as S = 10 + 10(10) + 10(10^2) + 10(10^3) + ... = 10 + 100 + 1000 + 10000 + ...

To find the sum of an infinite geometric series, we use the formula S = a/(1-r), where a is the first term and r is the ratio. So, for this series, the sum is S = 10/(1-10) = -10/9. However, this only represents the positive terms of the series. To get the full sum, we need to add the negative terms, which are represented by the same series but with a negative ratio. So, the full sum is S = -10/9 + (-10)/(-9) = -10/9 + 10/9 = 0. Therefore, 10+100+1000+10000+100000+...=-(1/9).

Why does this equation involve a negative sum?

The negative sum in this equation comes from the fact that the terms in the series alternate between positive and negative values. As the ratio is positive, the positive terms increase in value, while the negative terms decrease in value. This results in a net negative sum.

Can this equation be proven using different methods?

Yes, there are multiple ways to prove this equation. One method is to use the concept of limits in calculus. Another method is to use algebraic manipulation to find a closed form expression for the sum of the series, which will also result in the same answer of -(1/9).

How is this equation useful in mathematics?

This equation is useful in mathematics as it demonstrates the concept of infinite geometric series and how they can be summed using specific formulas. It also shows the importance of understanding the properties of infinite series and how they can be manipulated to find solutions.

Can this equation be applied to real-world situations?

While this equation may not have direct applications in real-world situations, it is still an important concept in mathematics. The idea of infinite series and their sums can be applied to various areas, such as finance, physics, and computer science. It also helps in understanding the concept of infinity and how it can be represented mathematically.

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