Question

Determine whether the series is cenverpent of diverpent. ∑ n=1/ n [(6.9) n−1 −(0.5) n ] cominigene tiversent at e in coaveraett, find its wam

Answer 1

The series is divergent based on the limit comparison test.

Step-by-step explanation:

The series ∑ n=1/n[(6.9)n−1 −(0.5)n] can be tested for convergence or divergence using the limit comparison test. We can compare it to the harmonic series ∑ n=1/n, which is known to diverge. Let's calculate the limit:

lim(n→∞) (n/n) * [(6.9)n−1 − (0.5)n]

Using L'Hôpital's rule, we can take the derivative of the numerator and the denominator to simplify the limit. After simplification, we get:

lim(n→∞) [(6.9)n * ln(6.9) - (0.5)n * ln(0.5)] / (1) = ∞

Since the limit is not a finite value (it goes to infinity), the series is divergent.