Final answer:
To determine the radius of convergence for the series with terms of the form ∑ a_n(x - c)^n where a_n = 6^n n^n and c = -7, the ratio test can be applied. The limit of the ratio of consecutive terms is calculated to find the radius.
Step-by-step explanation:
The given series seems to be a power series of the form ∑ a_n(x - c)^n where a_n = 6^n n^n and c = -7. To find the radius of convergence r for such series, one commonly uses the ratio test, which states that if lim(n->∞) |a_(n+1) / a_n|x < 1, the series converges. Therefore, we can find the radius of convergence by:
- Calculating the limit L = lim(n->∞) |a_(n+1) / a_n| for our specific a_n terms.
- Once we have L, we set L|x| < 1 and solve for |x|, which gives us the radius of convergence r.
In this particular case, due to the complexity of the series terms 6^n n^n, we might have to use additional techniques or make some approximations to compute this limit.