Evaluating Power Series and Finding the Interval of Convergence
The given power series is:
∑ (n=0 to ∞) (9^n)(x^n)
To find the interval of convergence, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges.
Let's apply the ratio test to our series:
lim (n→∞) |(9^(n+1))(x^(n+1)) / (9^n)(x^n)|
Simplifying the expression, we get:
lim (n→∞) |9x|
For the series to converge, |9x| < 1. Solving this inequality, we find:
-1 < 9x < 1
Dividing all terms by 9, we get:
-1/9 < x < 1/9
Therefore, the interval of convergence is (-1/9, 1/9).
It is important to check the convergence at the endpoints of the interval. Let's evaluate the series at x = -1/9 and x = 1/9.
When x = -1/9:
∑ (n=0 to ∞) (9^n)(-1/9)^n
= ∑ (n=0 to ∞) (-1)^n
This is an alternating series, and by the Alternating Series Test, it converges.
When x = 1/9:
∑ (n=0 to ∞) (9^n)(1/9)^n
= ∑ (n=0 to ∞) 1
This is a geometric series with a common ratio of 1, and by the Geometric Series Test, it diverges.
Therefore, the interval of convergence is (-1/9, 1/9], where the endpoint x = -1/9 is included.