Final answer:
To find the radius of convergence and interval of convergence of the series ∑n=1[infinity] n!(x−⁷ⁿ), we can apply the ratio test. The limit as n approaches infinity of |((n+1)!(x−⁷ⁿ₊₁))/(n!(x−⁷ⁿ))| is taken to determine the convergence of the series. Setting this limit equal to 1 and solving for x, we find that the series converges for all values of x, resulting in a radius of convergence of infinity and an interval of convergence of (-∞,∞).
Step-by-step explanation:
To find the radius of convergence and interval of convergence of the series ∑n=1[infinity] n!(x−⁷ⁿ), we can apply the ratio test. The ratio test states that if the limit as n approaches infinity of the absolute value of (aₙ₊₁/aₙ) is less than 1, then the series converges. Conversely, if the limit is greater than 1, then the series diverges. For our series, aₙ = n!(x−⁷ⁿ). Applying the ratio test, we have:
lim as n→inf |((n+1)!(x−⁷ⁿ₊₁))/(n!(x−⁷ⁿ))| < 1
Simplifying this expression, we get:
lim as n→inf |(n+1)(x−⁷ⁿ₊₁)/(x−⁷ⁿ)| < 1
Since this limit depends on the value of x, we can find the range of x values that make the limit less than 1 to determine the interval of convergence. The radius of convergence, denoted as R, is the distance from the center of the interval (x=0) to the first x value where the series either converges or diverges. Therefore, we need to find the values of x that make the limit equal to 1. Setting the limit equal to 1, we have:
lim as n→inf |(n+1)(x−⁷ⁿ₊₁)/(x−⁷ⁿ)| = 1
Simplifying this expression, we get:
lim as n→inf |(n+1)(x−⁷ⁿ₊₁)/(x−⁷ⁿ)| = lim as n→inf |x−⁷ⁿ₊₁|/|x−⁷ⁿ| = 1
Since the absolute value of a quotient is equal to the quotient of the absolute values, we can rewrite the expression as:
lim as n→inf |x−⁷ⁿ₊₁| = |x−⁷ⁿ|
Now, we can solve this equation for x:
|x−⁷ⁿ₊₁| = |x−⁷ⁿ|
This equation holds if the distance from x to ⁷ⁿ₊₁ is the same as the distance from x to ⁷ⁿ. In other words, x must be equidistant from ⁷ⁿ₊₁ and ⁷ⁿ. Since ⁷ⁿ₊₁ = ⁷ⁿ * ⁷, this condition can be expressed as:
|x−⁷ⁿ * ⁷| = |x−⁷ⁿ|
Using the fact that |ab| = |a|*|b|, we can rewrite the expression as:
|x−⁷ⁿ * ⁷| = |x−⁷ⁿ| ⟹ |x−⁷ⁿ|*|⁷| = |x−⁷ⁿ| ⟹ |x−⁷ⁿ|*7 = |x−⁷ⁿ|
Cancelling out the common factor of |x−⁷ⁿ|, we have:
7 = 1
Since 7 does not equal 1, this equation is not true for any value of x. Therefore, the limit as n approaches infinity of |(n+1)(x−⁷ⁿ₊₁)/(x−⁷ⁿ)| cannot equal 1, and the series converges for all values of x. Hence, the radius of convergence R is infinity, and the interval of convergence is (-∞,∞).