Final answer:
The question seems to have a typo but suggests finding a power series representation for a function centered at c=2 and determining its interval of convergence. A corrected assumption is that f(x) = 7 - x, which does not require a power series. Without the correct function, the interval of convergence can't be determined.
Step-by-step explanation:
To find a power series for the function f(x) = \frac{7-x}{1}, centered at c = 2, we assume that a typo occurred in the original question and the representation of the function should in fact be simply f(x) = 7 - x. Since this is a linear function, its power series representation is trivial. However, if you meant to write the function as a power series given by f(x) = \sum_{n=0}^{\infty}\frac{6^n + 1}{(x-2)^n}, we should look for a geometric series that can represent this function.
For the second part of the question, determining the interval of convergence for the power series, we need to apply the ratio test. Since there's a discrepancy in the given series (usually a geometric series has a common ratio, not an additive term in the numerator), it's not possible to provide a conclusive answer without the correct power series definition.
Note: The correct format for a power series is typically f(x) = \sum_{n=0}^{\infty} a_n (x - c)^n where a_n represents the coefficient for the n-th term and (x - c) is the term raised to the n-th power, with c being the center of the series.