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        riddles >> putnam exam (pure math) >> What is it?
        
(Message started by: Benoit_Mandelbrot on Apr 27^th, 2004, 6:05am)

Title: What is it?
Post by Benoit_Mandelbrot on Apr 27^th, 2004, 6:05am

Is this type of summation rational, purely irrational, or transcendental?

[sum]_i=1^[infty][a^{b^i}]

a,b[in][bbq]
a,b[ne]0,1
0<a<1
b>0

Title: Re: What is it?
Post by Pietro K.C. on Apr 28^th, 2004, 9:49am

Well, for a = 0 we have a 0 sum, so in general rational values are possible.

Nitpicking aside, if b [in] [bbn], this sum [hide]will always be irrational for a = 1/c for any positive integer c[/hide]. I'm not sure what to do with the general case yet, and I gotta go.

::
[hide]
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.
[/hide]
::

Title: Re: What is it?
Post by Pietro K.C. on Apr 30^th, 2004, 7:43pm

Some rather obvious results... (by the way, cool problem)

::
[hide]
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.
[/hide]
2. If a < 0 and b [notin] [bbz] I'm not sure what to do. What is (-1)^1/2?
[hide]
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.
[/hide]
::

Title: Re: What is it?
Post by Benoit_Mandelbrot on May 3^rd, 2004, 5:56am

I modified my post, so 0<a<1 and b>0 to eliminate problems. I have a thought that it is at least ::[hide] irrational, but I'm not sure if it is transcendental or not. [/hide]::
The summation can be read as 'a to the b to the i'.