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Under what conditions does an alternating series converge? An alternating series ∞ an n = 1 = ∞ (−1)n − 1bn, n = 1 where bn = |an|, converges if 0 < bn + 1 ≤ bn for all n.

Respuesta :

LudwigGR

Answer:

First, write the alternate series as [tex]\sum_{ n = 1 }^{\infty } (-1)^{n}  a_n[/tex] where the term [tex]a_n[/tex] is positive. From the Cauchy criterion we already know that [tex]a_n\rightarrow 0[/tex]. So, in order to  assure the convergence we only need to ask that the term  [tex]a_n[/tex] is monotonically decreasing.

Step-by-step explanation:

This is the Leibniz criterion for the convergence of alternate series.

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