Final answer:
To apply the ratio test to the series ∑ (4x) / n, calculate the limit of |a_(n+1) / a_n| as n approaches infinity. If |4x| < 1, the series converges; if |4x| > 1, it diverges; if |4x| = 1, the test is inconclusive.
Step-by-step explanation:
To determine the convergence of the series ∑ (4x) / n, we apply the ratio test. The ratio test compares the limit of the absolute value of the ratio of consecutive terms in the series.
Let's denote the nth term as a_n = (4x)^n / n. We need to find the limit of |a_(n+1) / a_n| as n approaches infinity:
Compute the ratio of the terms a_(n+1) and a_n:
- a_(n+1) = (4x)^(n+1) / (n+1)
- a_n = (4x)^n / n
- Divide a_(n+1) by a_n and take the absolute value:
- |a_(n+1) / a_n| = |(4x)^(n+1) / (n+1) * n / (4x)^n|
- |a_(n+1) / a_n| = |4x * n / (n+1)|
Take the limit as n approaches infinity:
lim (n → ∞) |4x * n / (n+1)| = |4x|
If this limit is less than 1, the series converges; if it is greater than 1, the series diverges.
If x is such that |4x| < 1, then the series converges. If |4x| > 1, then the series diverges. And, if |4x| = 1, the ratio test is inconclusive, and another test must be used.