Final answer:
To determine whether the series is convergent or divergent, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges, and if the limit is greater than 1, the series diverges. In this series, the nth term is given by T_n = 2^n / n^2, where n starts from 1. To apply the ratio test, we need to find the limit of (T_n+1 / T_n) as n approaches infinity. Calculating the limit, we find that it is equal to 2, which is greater than 1, indicating that the series diverges.
Step-by-step explanation:
To determine whether the series is convergent or divergent, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges, and if the limit is greater than 1, the series diverges.
In this series, the nth term is given by T_n = 2^n / n^2, where n starts from 1. To apply the ratio test, we need to find the limit of (T_n+1 / T_n) as n approaches infinity.
Calculating the limit, we have:
lim(n->infinity) (T_n+1 / T_n) = lim(n->infinity) [(2^(n+1) / (n+1)^2) / (2^n / n^2)]
Using properties of exponents and simplifying, we get:
lim(n->infinity) (T_n+1 / T_n) = lim(n->infinity) (2 / (1 + 1/n)^2) = 2
Since the limit is equal to 2, which is greater than 1, the series diverges.