Final Answer:
The interval of convergence for the series is the empty set: Ø, meaning the series diverges for all values of x.
Step-by-step explanation:
Interval of Convergence for the Series Represented by f(x) = 7/6 x⁶
We're tasked with determining the interval of convergence for the series generated by integrating the function f(x) = 7/6 x⁶.
Approach:
The ratio test is employed to determine the interval of convergence for the series. The ratio test asserts that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges. The test is inconclusive if the limit is equal to 1. The series diverges if the limit is greater than 1.
Step 1: Identify the Series
The series generated by integrating the function f(x) = 7/6 x⁶ is:
1/6x⁷ + C + 1/6x¹⁴ + C + 1/6x²¹ + C + ...
where C is an arbitrary constant of integration.
Step 2: Determine the Ratio of Successive Terms
The ratio of successive terms is:
(1/6x¹⁴ + C) / (1/6x⁷ + C) = x⁷ / 1
Step 3: Evaluate the Limit of the Ratio
The limit of the absolute value of the ratio as x approaches positive and negative infinity is:
lim_{x→∞} |x⁷ / 1| = ∞
lim_{x→−∞} |x⁷ / 1| = ∞
Conclusion:
Since the limit is greater than 1 for all values of x, the series diverges for all values of x. Therefore, the interval of convergence is the empty set: Ø.