Final answer:
The series Σ_{n=1}/[infinity] ∫{(x-4)ⁿ{n} converges for values of x between 3 and 5.
Step-by-step explanation:
The given series is Σ_{n=1}/[infinity] ∫{(x-4)ⁿ{n}
To determine the values of x for which the series converges, we need to find the values of x that make the common term of the series approach 0 as n approaches infinity.
Since (x-4)ⁿ{n} is the common term, we can see that for the series to converge, the absolute value of (x-4) must be less than 1, which means -1 < x-4 < 1.
Simplifying the inequality, we get 3 < x < 5.
Therefore, the series Σ_{n=1}/[infinity] ∫{(x-4)ⁿ{n} converges for values of x between 3 and 5.