Final answer:
To determine the value of k that makes the series converge, apply the ratio test. The series converges for any value of k.
Step-by-step explanation:
To determine the value of k that makes the series converge, we need to apply the ratio test. The ratio test states that if the limit as n approaches infinity of the absolute value of the ratio between the (n + 1)-th term and the n-th term is less than 1, then the series converges. In this case, the series is given by ∑[infinity]n=1 7ⁿ/nᵏ.
Applying the ratio test, we have:
limn→∞ |(7ⁿ⁺¹)/((n + 1)ᵏ)/(7ⁿ)/(nᵏ)| < 1
limn→∞ |7/(n + 1)|ⁿ⁺¹|nⁿ| = 7 × limn→∞ |n + 1|ⁿ⁺¹|nⁿ| = 0
Since the limit of the expression is equal to 0, the series converges for any value of k.