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I read in Rudin's Analysis that sequence 1/n failes to converge in the set of positive real numbers. How comes?
HallsofIvy said:There is a theorem (the "r test") that says that [tex]\sum n^{r}[/tex] converges if and only if r< 1 (or that [itex]\sum 1/n^r[/itex] converges if and only if r> 1.)
So "[itex]\sum 1/n[/itex]" is a borderline case: it diverges by the integral test:
[itex]\int_1^\infty dx/x[/itex] does not converge so the series does not converge.
The sequence 1/n not convergent refers to a mathematical sequence where the values of the terms become closer and closer to 0, but never actually reach 0. This means that the sequence does not have a limit, and therefore is not convergent.
The sequence 1/n is not convergent because as n approaches infinity, the terms in the sequence become infinitely small, but never actually reach 0. Therefore, the sequence does not have a limit and is not convergent.
A convergent sequence has a limit, meaning that as the values of the terms get closer and closer together, they eventually reach a specific value. A non-convergent sequence, on the other hand, does not have a limit and the values of the terms do not approach a specific value, even as the terms get closer and closer together.
Non-convergent sequences have several applications in fields such as physics, engineering, and economics. For example, they can be used to model population growth, financial investments, and the behavior of electrical circuits.
To determine if a sequence is convergent or non-convergent, you can use various tests such as the divergence test, comparison test, and ratio test. These tests evaluate the behavior of the terms in the sequence and can determine if the sequence has a limit or not.