Most of us are aware of the following consequence of Fundamental Theorem of Arithmetic:
There are infinitely many prime numbers.
The classic proof by Euclid is easy to follow. But I wanted to share the following two analytic equivalents (infinite series and infinite products) of the above purely arithmetical statement:
For proof, refer to this discussion: https://math.stackexchange.com/q/361308/214604
- , where is any complex number with .
The outline of proof, when is a real number, has been discussed here: http://mathworld.wolfram.com/EulerProduct.html
While reading Lillian Lieber’s book on infinity, I came across an astonishing example of infinite set (on pp. 207). Let’s call the property of existence of a rational number between given two rational number to be “beauty” (a random word introduced by me to make arguments clearer).
The set of rational numbers between 0 and 1 are arranged in ascending order of magnitude, and all of them are coloured blue. This is clearly a beautiful set. Then another another set of rational numbers between 0 and 1 is taken and arrange in ascending order of magnitude, but all of them are coloured red. This is also a beautiful set. Now, put these two sets together in such a way that each blue number is immediately followed by the corresponding red number. For example, 1/2 is immediately followed by 1/2 etc. It appears that if we interlace two beautiful sets, the resulting set should be even more beautiful. But since each blue number has an immediate successor, namely the corresponding red number, so that between these two we can’t find even a single other rational number, red or blue, the resulting set is NOT beautiful.
The set created above is called Huntington’s Red-Blue set. It is an ingenious invention, where two beautiful sets combined together lead to loss of beauty. For more details, read the original paper:
Huntington, Edward V. “The Continuum as A Type of Order: An Exposition of the Modern Theory.” Annals of Mathematics, Second Series, 7, no. 1 (1905): 15-43. doi:10.2307/1967192.
I ended up with a task at the end of my post last month: Resting with ‘Nested Radicals’
I have figured out that is a diverging series. But now I am struck with another problem:
What is the corresponding sequence for nested radical series?
In general, I am given a sequence and I write the corresponding series and check its convergence/divergence. But what if I am given a series without a general term (though wrong in many cases, as a given pattern of finite terms may be written as many different general terms.). For nested radicals I believe that there is only one general term and without this general term I will not be able to write terms of corresponding sequence (as in case of nested radicals I simply get partial sums). So I have a bigger question to ask:
Does there exists a sequence corresponding to any given series?