This is a very controversial sum
What is the sum of all perfect powers of 2 that are greater than 1?
Details and Assumptions
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Read my solution if you are confused.
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Assume that the sum has a real value, and treat it as such.
The answer is -2.
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2 solutions
Your logical fallacy is the assumption that X has a defined value. The "value" of x, which is theoretically undefined, can be said to approach infinity. Thus, the value of X/2 also approaches infinity. The operation of subtraction is undefined for undefined values such as infinity, where your logical fallacy lies.
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Can someone from the Brilliant community please back me up/correct me on this?
Ah, but I just proved that X does have a value. And I don't want to have an argument, but if you want to, here's the starting lineup:
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My side : Ramanujan, Grandi, Euler, and the general public.
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Your side : Other Brilliant members.
The same type of techniques have been used like the names above and are widely accepted.
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You have not proven that X does have a value. All that you have shown, is that if X does have a value, then it's possible for the value to be 1. You 'lost' the solution of infinity, because you committed the error of ∞ − ∞ which is undefined. This is similar to proofs which purport to show that − 1 = 1 .
There are possible interpretations of the infinite sum which yield a value that is not infinity. However, you need to be explicit about the model that you're working in.
If I asked a question of "What is
1
+
2
?", and gave an answer of "10" (Oh, because I am working in ternary, I just forgot to tell you) or "0" (Oh, because I was working in modulo 3, it's obvious in the question), then my question would not be clear.
As such, I've added "This problem is not clearly phrased" to the question.
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@Calvin Lin – Yeah, another thing I did not understand when I was trying the problem was the term "perfect". I assumed that Finn meant all the perfect numbers that are also powers of 2, in which case the answer is just two.
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@Tanishq Aggarwal – That's a proper term, to strictly ensure that it is of the form 2 n , where n is an integer. It just means 'power'.
You've seen this before, where we interchangeably use squares and perfect squares to both mean n 2 , where n is an integer. Sometimes, squares can mean x 2 , where x need not be an integer.
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@Calvin Lin – Ah oops. Guess that was dumb of me.
But then X divided by 2 would make x/2 bigger than x, which does not make sense.
Well...what you have just done is that you have used the formula for summation of an infinite GP.... S = 1 − r a . the formula is valid only if ∣ r ∣ < 1 ..... and here r = 2 thats why you are getting a weird answer
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No, that's actually not what I used. But it does produce the desired result, and thus -2 is correct by using your formula or my algebra.
Hello Finn Hulse..Your solution is indeed correct in a way of solving equations but does not abide by the mathematical logic..Similarly there is a problem..I once liked when I was a child.. If a=10,b=10 ,,then prove a+b =10 which leaves behind all the logic of Mathematics and gives u a brilliant solution. But that doesn't mean it is correct.But I appreciate your solution ... :)
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If you can prove me wrong, I will more readily accept defeat. There is no flaw in my algebra, like other trick solutions to prove false statements. I don't divide by zero, I don't ignore negative square-roots, etc..
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..but you do subtract infinity-infinity form....
how com ethat the some of positive numbers results in a negative one ?? strange !!! isn't it ???
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Oh, it's strange alright.
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Once again, let me remind you that this strange result is a result of the logical fallacy I mentioned above. Also, can you cite proof that Ramanujan, Euler, and Grandi are on "your side"? That seems rather intriguing...
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@Tanishq Aggarwal – Okay, look up "1+2+3+4..." in Wikipedia, and read the article. Also, look up "Grandi's Series", and "Ramanujan's method for finding 1-2+3-4+5...". Also, in that article first mentioned, read the paragraph of how Euler proved Grandi's series using a similar method to mine for which I proved my problem correct.
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@Finn Hulse – I am on both of your sides.There is sense in thinking that the sum is infinity, but that's just too boring :D.Same is with ζ ( − 1 ) .But I agree that the sum has no value in the usual sense, but a value of it can be derived by violating general rules.
hey ........... I acknowledge that ur solution is a pretty good one......... but for logic's sake can ever the sum of positive nos. be negative..............?????????/////////
Similar solution to yours, Finn.
( 2 + 4 + 8 + . . . ) equals to ( 2 − 1 ) ( 2 + 4 + 8 + . . . ) . If you simplify this you get ( 4 + 8 + 1 6 + . . . ) − ( 2 + 4 + 8 + 1 6 . . . ) . Cancelling out the equivalent numbers, you get an answer of − 2 .
Awesome job! Pretty cool, huh?
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Yeah. Mathematical outlaws do this.
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I'm on the run from the law baby... The Law of Large Numbers. ;)
If you are reading this, chances are, you are a very frustrated person. You have tried all sorts of strange numbers, and not one of them as been right... probably. Here is a sound proof that I think I discovered that will show you why the answer is indeed -2. Let's let X = 2 + 4 + 8 + 1 6 + 3 2 . . . . If we divide by 2, we get X / 2 = 1 + 2 + 4 + 8 + 1 6 + 3 2 . . . . But wait a second. Look at all of the terms except for the 1 in the X / 2 equation. That's just equal to X ! We can set up an equation! X / 2 = 1 + X . Solving this, we find that X = − 2 . If you can find a single fault in my logic, I will regard you as the smartest person in the world. :D