Answer:
0
Convergent
Explanation:
aₙ = 2ⁿ / n!
aₙ₊₁ = 2ⁿ⁺¹ / (n+1)!
lim(n→∞)│(2ⁿ⁺¹ / (n+1)!) / (2ⁿ / n!)│
lim(n→∞)│(2ⁿ⁺¹ / (n+1)!) × (n! / 2ⁿ)│
lim(n→∞)│(2ⁿ⁺¹ / 2ⁿ) × (n! / (n+1)!)│
Notice that (n+1)! = (n+1) n!. So this reduces to:
lim(n→∞)│2 × 1 / (n+1)│
0
The limit is less than 1, so the series is absolutely convergent.