Final answer:
The Taylor series for f(x) = 6/x centered at c = 1 is the infinite sum of terms (-1)^n * 6 * (x - 1)^n / n!, derived by evaluating the successive derivatives of the function at x = 1.
Step-by-step explanation:
The student has asked to find the Taylor series for the function f(x) = 6/x, centered at c = 1, which is represented by the notation f(x) = ∑ₙ=0[infinity]. To find a Taylor series, we need to calculate the derivatives of f(x) evaluated at c and then construct the series.
The zeroth derivative of f(x) (which is just f(x) itself) evaluated at c = 1 is f(1) = 6. The n-th derivative of f(x) at x = 1 is f(n)(1) = (-1)n × n! × 6, because the derivative of 6/x produces a factorial pattern when evaluated at x = 1. Thus, the Taylor series for f(x) around c = 1 is the sum of terms (-1)n6(x - 1)n/n!.