Another necessary condition

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In summary, the Cauchy definition of convergence of a series involves the condition that the partial sums of the series approach a limit as the number of terms increases. This can be rewritten in terms of a positive decreasing sequence, and taking the limit can provide another necessary criterion for convergence. However, it is difficult to generalize this criterion to series with different types of terms, as seen with the example of the alternating harmonic series. While it may be possible to extend this criterion to positive term series, the proof remains elusive.
  • #1
Starting from the Cauchy definition of convergence of a series :

[tex]\forall N,\epsilon>0,\exists N_0 | k>N_0\Rightarrow |\underbrace{\sum_{n=1}^{N+k}u_n-\sum_{n=1}^k u_n}_A |<\epsilon[/tex]

rewriting A in terms and considering a positive decreasing sequence :

[tex]A\Rightarrow \epsilon>u_{N+k}+\ldots +u_{k+1}>Nu_{N+k}[/tex]

one finds by taking the limit another necessary criterion :

[tex]\lim_{n\to\infty}n u_n=0[/tex].

This implies for example that the harmonic series cannot converge.

My question is why we don't see this at school but only the condition [tex]u_n\to 0[/tex] ?
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  • #2
You are imposing the additional restriction that you have a positive decreasing sequence. The "Divergence Test" that is taught in school applies to all series.
  • #3
Do you think it's possible to generalize this to any kind of series ?
  • #4
jk22 said:
Do you think it's possible to generalize this to any kind of series ?
Take the alternating harmonic series.
It converges.
But, it wouldn't, according to the simplest form of generalization of your criterion.
The upshot is that it probably is VERY difficult to generalize this Cauchy criterion into an independent necessary criterion for convergence for series of a more general type than those with positive, decreasing terms.
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  • #5
I thought maybe it is possible to generalize to positive term series (not forcedly decreasing).

The sum of the terms then becomes : [tex]Nu_m[/tex] where u_m is the minimum of the terms between N+k and k, but then remains to prove that m tends towards N when N tends to infinity, but I have no idea how to do that.

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