Alternating Series Test for these questions?

The Alternating Series Test is not too suited for these series given...
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1) Note that cos(nπ) = (-1)^n for any integer n;
so the series equals Σ(n = 1 to ∞) 1/n^5, which is a convergent p-series.
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2) Divergent by the Ratio Test, since r = ∞ > 1:
r = lim(n→∞) |[(-1)^(n+1) (n+1)! / e^(n+1)] / [(-1)^n n!/e^n]|
..= lim(n→∞) (1/e) (n+1)
..= ∞.
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3) Convergent by the Ratio Test, since r = 1/e < 1:
r = lim(n→∞) |[(-1)^(n+1) (n+1)! / (n+1)^(n+1)] / [(-1)^n n!/n^n]|
..= lim(n→∞) n^n / (n+1)^n
..= lim(n→∞) 1 / [(n+1)^n/n^n]
..= lim(n→∞) 1/(1 + 1/n)^n
..= 1/e, by the limit definition of e
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4) Divergent by the Ratio Test, since r = e > 1:
r = lim(n→∞) |[(-1)^(n+1) (n+1)^(n+1)/(n+1)!] / [(-1)^n n^n/n!]|
..= lim(n→∞) (n+1)^n/n^n
..= lim(n→∞) (1 + 1/n)^n
..= e, by the limit definition of e
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5) Convergent by the Ratio Test, since r = 0 < 1:
r = lim(n→∞) |[(-1)^(n+1) e^(n+1)/(n+1)!] / [(-1)^n e^n/n!]|
..= lim(n→∞) e/(n+1)
..= 0.
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I hope this helps!