- #1
sutupidmath
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we have to prove that
10+100+1000+10000+100000+...=-(1/9)
any ideas?
10+100+1000+10000+100000+...=-(1/9)
any ideas?
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.
HallsofIvy said:I would have thought it obvious from the start that a sum of positive numbers cannot be negative!
!
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?
Yes, it diverges.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?
the error was when i let S = 10 + 100 + 1000 + ...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 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).
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.
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).
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.
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.