Given our function f(x) as a power series:
f(x) = 7 + 3x - 4x^2 + 7x^3 + 3x^4 - 4x^5 + ...
We are told the coefficients follow a pattern where every third coefficient (starting from 0) is 7, then 3, then -4, respectively.
To find the interval of convergence, we can conduct the ratio test, which states that a series ∑ a_n * x^n converges absolutely if the limit as n approaches infinity of the absolute value of (a_(n+1) * x^n+1) / (a_n * x^n) is less than 1.
Due to the observed pattern in the coefficients, we can say that this series appears to converge for -1 < x < 1 because the coefficients do not grow too rapidly (remains 7, 3, -4 for all n).
Therefore, the interval of convergence for the given series is (-1, 1).
Next, we want to find the closed-form formula.
This is a bit tricky because the closed-form depends on the value of the input modulo 3.
From the given pattern of coefficients, we can write f(x) as a piecewise function where:
- If floor(x) modulo 3 = 0, then the output is 7.
- If floor(x) modulo 3 = 1, then the output is 3.
- If floor(x) modulo 3 = 2, then the output is -4.
Expressing this as a sum from 0 to infinity, we have:
f(x) = ∑ [(7 if floor(x) mod 3 = 0),
(3 if floor(x) mod 3 = 1),
(-4 if floor(x) mod 3 = 2)],
(where the sum goes from x=0 to infinity).
However, due to the alternating nature of the series (because of mod operations) and the limitations of numerical calculations, we may not find a numerical closed-form expression, thus yielding an undefined result, represented as 'nan' in computing terms.
In conclusion, we have found that our function f(x) converges for x values in the open interval (-1, 1), but does not have a typical closed-form solution that can be expressed in a finite manner due to its alternating behavior.