Examine the convergence of the series Σn=0∞ sin(nx)(n+1)(n+2).

If f: [-1, 1] → ℝ is continuous such that ∫-11 f(t) dt = 1, then find limn → ∞ ∫-11 f(t) cos2(nt) dt.

Let fn(x) = (2x)n, x ∈ [0, 12] and n ∈ ℕ be a sequence of functions; then _________.

Let a > 1 and f: (a, ∞) → ℝ be defined by f(x) = 1/x4 if x ∈ ℚc ∩ [a, ∞) and f(x) = 0 otherwise. Then the improper integral ∫a∞ f(x) dx will be _________.