Special cases - Alternating Series Test?

Can an alternating series be convergent even if it doesn't satisfy all the conditions of the alternating series test? like it just satisfied the first or second condition. Or fail to satisfy both conditions but still converges?


Or will it be divergent even if it satisfy both conditions?


Give an example for each special case.Special cases - Alternating Series Test?
Every convergent series must satisfy the condition that the limit of the terms is 0. (Compare this to the divergence test.)





It certainly is possible to have an alternating series where the absolute value of the terms is not decreasing but nevertheless the series converges. For example, consider the sequence with terms An = 1/n² for n even and An = -1/n³ for n odd. However, it is possible without this criterion for a sequence to diverge: e.g. An = 1/n for n even and An = -1/n² for n odd diverges. (I leave the proof of this as an exercise.)





Obviously, any alternating series that meets the two conditions will converge; that is what the theorem is all about!





So, in summary:


● Any series where the limit of the terms is not 0 diverges


● A series where the limit of the terms is 0 but where the absolute value of the terms is not decreasing may or may not converge.


● Any alternating series where the limit of the terms is 0 AND where the absolute value of the terms forms a decreasing sequence, is a convergent series.





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