Find the radius of convergence, R,and the interval, I, of the series.
starting n=1
[(x-4)^n]/(n^n)
To find the radius of convergence, R, and the interval, I, of the series, we can use the ratio test.
The ratio test states that for a series ∑aₙ, if the limit of the absolute value of the ratio of consecutive terms is L, then the series converges if L < 1, and diverges if L > 1.
For our series, the general term aₙ is given by [(x-4)^n]/(n^n).
Now, let's apply the ratio test. Taking the limit of the absolute value of the ratio of consecutive terms, we have:
lim (n→∞) |((x-4)^(n+1))/(n+1)^(n+1)| / |((x-4)^n)/n^n|
Simplifying, we get:
lim (n→∞) |(x-4)^(n+1) * n^n| / |(x-4)^n * (n+1)^(n+1)|
Further simplifying, we have:
lim (n→∞) |(x-4)^(n+1) * n^n| / |(x-4)^n| * |(n+1)^(n+1)|
Next, we can simplify the expression inside the limit by dividing the absolute value terms and rearranging, resulting in:
lim (n→∞) |((x-4) * n / (n+1))^(n+1)| / |(n/n^n)|
Now, as n approaches infinity, the expression (x-4) * n / (n+1) approaches (x-4). Additionally, we know that lim (n→∞) (n/n^n) = 0.
Therefore, our limit simplifies to:
lim (n→∞) |(x-4)^(n+1) / n^(n+1) * (n+1)/n)|
= |(x-4)| lim (n→∞) ((n+1)/n)
= |(x-4)|
To ensure the series converges, we must have |(x-4)| < 1. This implies that the series converges for values of x within a distance of 1 from 4.
Thus, the radius of convergence, R, is 1, and the interval, I, is (3, 5).