Final answer:
The series ∑n=1∞(an) is divergent, while the sequence {an} is convergent.
Step-by-step explanation:
The given sequence is defined as an = (256n) / (242n). To determine the convergence of the series ∑n=1∞(an), 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, then the series converges. Let's apply the ratio test to the given series:
lim(n→∞) |(an+1 / an)| = lim(n→∞) |(256(n+1) / 242(n+1)) / (256n / 242n)|
Simplifying this expression, we get:
lim(n→∞) ((n+1) / n)
As n approaches infinity, the limit of ((n+1) / n) is equal to 1. Since 1 is less than 1, the series ∑n=1∞(an) is divergent.
To determine the convergence of the sequence {an}, we need to analyze the behavior of the individual terms. Since an = (256n) / (242n) can be simplified to an = 1.0566, the sequence {an} is a constant sequence with the value 1.0566. Therefore, the sequence {an} is convergent.