For any sequence s_n consider the arithmetic mean
t_n = (s1 + s2 + ... + sn) / n
Prove sn -> s implies tn -> s. Prove there are divergent sequences s_n which in this manner give rise to convergent sequences t_n.
If we can set each where . Now is...
(1)
... and because each term of the sum at the second term of (1) tends to 0 the sum itself tends to 0...
An example of divergent series that produces a convergent sequence is the armonic series. Ibn effect is...
(2)
... where is the 'Euler's constant'. From (2) You immediately derive that ...
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