- #1

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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] ?

[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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