answer:
To find the interval of convergence of the given power series ∑ (20)(9x)^n, we can use the ratio test.
The ratio test states that for a power series ∑ aₙ(x - c)^n, if the limit as n approaches infinity of |aₙ₊₁ / aₙ * (x - c)| is less than 1, then the series converges absolutely.
In this case, aₙ = (20)(9x)^n. To apply the ratio test, we need to find the limit of |aₙ₊₁ / aₙ * (x - c)| as n approaches infinity:
|[(20)(9x)^(n+1)] / [(20)(9x)^n * (x - c)]|
Simplifying this expression, we have:
|(9x)(x - c) / (9x)|
The factor of 9x cancels out, leaving:
|x - c|
For the series to converge absolutely, |x - c| must be less than 1.
Now, let's consider the endpoints of the interval. When |x - c| = 1, the series may or may not converge. To determine this, we can substitute x = c + 1 and x = c - 1 into the original series and check for convergence.
When x = c + 1, the series becomes:
∑ (20)(9(c+1))^n = ∑ (20)(9c+9)^n
Since the common ratio 9c + 9 is not less than 1, the series does not converge.
When x = c - 1, the series becomes:
∑ (20)(9(c-1))^n = ∑ (20)(9c-9)^n
Again, since the common ratio 9c - 9 is not less than 1, the series does not converge.
Therefore, the interval of convergence is |x - c| < 1, or in interval notation, (c - 1, c + 1).
Note: In this case, we don't have a specific value for c given, so we cannot determine the exact interval. However, we know that the interval lies between (c - 1, c + 1).