To find the interval of convergence of the power series with the general term (-1)^(n+1) * (x-8)^n / (n^9), we start by using the ratio test. The ratio test states that a power series converges if the limit as n approaches infinity of the absolute value of the (n+1)th term divided by the nth term is less than 1.
Our general term here is (-1)^(n+1) * (x-8)^n / (n^9). We apply the ratio test to this term. That means, we replace n with (n+1) in our general term and divide it by the original general term.
Now we simplify this ratio. The result is the absolute value of (x - 8).
Next, we need to evaluate the limit of this ratio as n approaches infinity. For our case, the limit of the ratio is the absolute value of (x - 8), which does not depend on n and hence does not vary with n. Therefore, the limit as n approaches infinity is the same expression, the absolute value of (x - 8).
The ratio test states that the power series converges where the limit of the ratio is less than 1. Hence, we can choose this limit as < 1 for the interval of convergence.
In other words, the power series converges for values of x that make the absolute value of (x - 8) less than 1.
Solving this inequality, we find that the interval of convergence of the power series is (7, 9). It is important to note that this interval is open, which implies that the series does not necessarily converge at the endpoints x=7 and x=9.
So, the interval of convergence of the given power series is (7, 9).