Question

Determine whether the sequence converges or diverges. If it converges, find the limit. [Hint:

Answer 1

(a) converges; consider the function f(x) = a ˣ, which converges to 0 as x gets large for |a | < 1. Then the limit is 2.

(b) converges; we have

4ⁿ / (1 + 9ⁿ) = (4ⁿ/9ⁿ) / (1/9ⁿ + 9ⁿ/9ⁿ) = (4/9)ⁿ × 1/(1/9ⁿ + 1)

As n gets large, the exponential terms vanish; both (4/9)ⁿ → 0 and 1/9ⁿ → 0, so the limit is 1.

(c) converges; we know ln(n ) → ∞ and arctan(n ) → π/2 as n → ∞. So the limit is π/2.