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Ed Quanta
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Why does the series of 1/n! diverge in the p-adic metric?In other words, how do I show that the lim of 1/n! (in the p-adic metric) does not equal 0 because it is >1
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No, but I am not sure how to show this. I know that every n has a prime factorization, and that as n increases, p will go into n more and more times. But what does this mean?
Ed Quanta said:The only answer to that question that makes sense to me is 1
Divergence in p-adic metric refers to the behavior of a sequence in the p-adic metric space. A sequence is said to diverge if it does not have a limit in the p-adic metric space.
The concept of divergence in p-adic metric is unique because the p-adic metric space has different properties compared to other metric spaces. In p-adic metric, the distance between two numbers is determined by the highest power of p that divides their difference. This leads to different behavior of sequences and their convergence or divergence.
A sequence in p-adic metric diverges if its terms become arbitrarily large in the p-adic metric space. This can happen if the sequence has a term with a high power of p, which dominates the other terms and makes the sequence diverge.
Yes, a sequence can diverge in p-adic metric but converge in other metric spaces. This is because the behavior of sequences in p-adic metric is based on the properties of p-adic numbers, which are different from real or complex numbers in other metric spaces.
Studying divergence in p-adic metric has applications in number theory, algebraic geometry, and cryptography. It also has connections to other areas of mathematics, such as fractal geometry and dynamical systems.