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Topic: What is it? (Read 615 times) |
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Benoit_Mandelbrot
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What is it?
« on: Apr 27th, 2004, 6:05am » |
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Is this type of summation rational, purely irrational, or transcendental?
[sum]i=1[infty][ab^i]
a,b[in][bbq]
a,b[ne]0,1
0<a<1
b>0
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| « Last Edit: May 3rd, 2004, 5:58am by Benoit_Mandelbrot » |
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Because of modulo, different bases, and significant digits, all numbers equal each other!
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Pietro K.C.
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Re: What is it?
« Reply #1 on: Apr 28th, 2004, 9:49am » |
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Well, for a = 0 we have a 0 sum, so in general rational values are possible.
Nitpicking aside, if b [in] [bbn], this sum will always be irrational for a = 1/c for any positive integer c. I'm not sure what to do with the general case yet, and I gotta go.
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To see this, consider the c-ary expansion of the partial sums. Clearly, there will be a digit 1 at every bi-th position after the dot, and zeros elsewhere. Hence this expansion is not periodic, and the limit cannot be rational.
I leave it as an exercise to the reader to make the above 'proof' precise, because I got a class right now.
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| « Last Edit: Apr 28th, 2004, 9:51am by Pietro K.C. » |
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"I always wondered about the meaning of life. So I looked it up in the dictionary under 'L' and there it was --- the meaning of life. It was not what I expected." (Dogbert)
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Pietro K.C.
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Re: What is it?
« Reply #2 on: Apr 30th, 2004, 7:43pm » |
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Some rather obvious results... (by the way, cool problem)
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1. If a > 1 the sum will not converge. If b [ge] 0, then bi goes to either 0 or [infty]; hence the terms of the series go to 1 or [infty]. Since the terms of a convergent series must go to 0, the limit will not exist. Even if b < 0, the subsequence b2i still behaves exactly as before, and the general term does not go to 0.
2. If a < 0 and b [notin] [bbz] I'm not sure what to do. What is (-1)1/2?
3. If a < -1 and b [in] [bbz] we still have the problem of b2i going to [infty], and the general term not going to 0.
4. If -1 < a < 1 and b < 0 we have b2i+1 < 0, therefore the odd-numbered terms don't go to 0.
5. If 0 < a < 1 and 0 < b < 1, lim bi = 0; I'm getting sick of this goshdarned general term not going to zero.
6. If -1 < a < 1 and b [in] [bbz] is greater than 1, then the series converges (at last!), by a simple comparison to the geometric series.
7. If 0 < a < 1 and b > 1 but not necessarily an integer, the sum still converges by a more complicated comparison to terms 'far enough' into the geometric series. Think of how bi grows, compared to i, for large enough i.
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| « Last Edit: Apr 30th, 2004, 7:58pm by Pietro K.C. » |
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"I always wondered about the meaning of life. So I looked it up in the dictionary under 'L' and there it was --- the meaning of life. It was not what I expected." (Dogbert)
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Benoit_Mandelbrot
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Re: What is it?
« Reply #3 on: May 3rd, 2004, 5:56am » |
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I modified my post, so 0<a<1 and b>0 to eliminate problems. I have a thought that it is at least :: irrational, but I'm not sure if it is transcendental or not. ::
The summation can be read as 'a to the b to the i'.
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| « Last Edit: May 3rd, 2004, 5:58am by Benoit_Mandelbrot » |
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Because of modulo, different bases, and significant digits, all numbers equal each other!
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