determine the radius of convergence of the power series:
Σ(n=1 to inf) [(2^n * n! * x^n)/ n^n]
To determine the radius of convergence, we can make use of the Ratio Test. The Ratio Test states that for a power series Σaₙxⁿ, the series converges if the limit of the ratio of consecutive terms |aₙ₊₁/aₙ| as n approaches infinity is smaller than 1, and diverges if the limit is larger than 1.
Let's apply the Ratio Test to the given series:
|aₙ₊₁ / aₙ| = |[(2^(n+1) * (n+1)! * x^(n+1)) / (n+1)^(n+1)] / [(2^n * n! * x^n) / n^n]|
= |[(2^(n+1) * (n+1)! * x^(n+1)) / (n+1)^(n+1)] * [(n^n) / (2^n * n! * x^n)]|
Next, let's simplify the expression by canceling out common terms:
|aₙ₊₁ / aₙ| = |[(2 * (n+1) * x^(n+1)) / (n+1)^(n+1)] * [(n^n) / (2^n * x^n)]|
Simplifying further:
|aₙ₊₁ / aₙ| = |[2 * x^(n+1) / (n+1)] * [(n^n) / (2^n * x^n)]|
Remove the absolute value brackets:
|aₙ₊₁ / aₙ| = 2 * |x^(n+1) / (n+1)| * |n^n / (2^n * x^n)|
Now, let's take the limit of this expression as n approaches infinity.
Taking the limit of |x^(n+1) / (n+1)| as n approaches infinity, the term x^(n+1) will vanish since it approaches zero faster than the denominator (n+1) grows.
Limiting |n^n / (2^n * x^n)| as n approaches infinity, we can rewrite it as [(n/x)^n / 2^n]. The term (n/x)^n approaches infinity, while 2^n remains constant.
Thus, we can conclude that the limit of |aₙ₊₁ / aₙ| as n approaches infinity is 0. Since 0 is less than 1, the series converges for all x values. Therefore, the radius of convergence is infinity.
In other words, the power series Σ[(2^n * n! * x^n)/ n^n] converges for all values of x.