1. contradictions in convergence and divergence rules. need clarification

hello, so i'm reviewing the chapter on absolute and conditional divergence, and this chapter has me completely baffled.

are p-series not always applicable?

because, for example,

infinite
sigma [(-1)^n]/(n)^1/2
n=1

ace of n = 1/n^(1/2)

converges, even though p=1/2 < 1.

but at the same time,

infinite
sigma 1/n^(1/2)
n=1

is divergent because the p-series somehow applies to this one and not that one.

are p-series only applicable for non-alternating series? when does the leibniz rule override the p-series rule?

also, for

infinite
sigma [(-1)^n]/[(n^2)+1]^1/2
n=1

converges. does that mean

infinite
sigma 1/[(n^2)+1]^1/2
n=1

converges?

andy why is ace of n called a sequence that converges to zero when it is NOT alternating and is therefore supposed to diverge?

i'm very confused.

2. Originally Posted by wyhwang7
hello, so i'm reviewing the chapter on absolute and conditional divergence, and this chapter has me completely baffled.

are p-series not always applicable?

because, for example,

infinite
sigma [(-1)^n]/(n)^1/2
n=1

ace of n = 1/n^(1/2)

converges, even though p=1/2 < 1.

but at the same time,

infinite
sigma 1/n^(1/2)
n=1

is divergent because the p-series somehow applies to this one and not that one.

are p-series only applicable for non-alternating series? when does the leibniz rule override the p-series rule?
Yes. If you check your text you should see that a "p" test is that all the terms be postive. If by the "Leigniz" rule, you mean that "$\displaystyle \sum(-1)^na_n$ converges whenever $\displaystyle a_n$ goes to 0", it doesn't have to "override" the p-test because it only applies to alternating series while the p-test only applies to positive series (if all terms are negative you can factor out -1 so more generally it applies to series where the terms are all of the same sign).

also, for

infinite
sigma [(-1)^n]/[(n^2)+1]^1/2
n=1

converges. does that mean

infinite
sigma 1/[(n^2)+1]^1/2
n=1

converges?
No it doesn't.

andy why is ace of n called a sequence that converges to zero when it is NOT alternating and is therefore supposed to diverge?

i'm very confused.
A sequence and a series are very different things. Nor is it true that any series that is not alternating diverges.