Problem 27 - Calculus Early Transcendentals

Question

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{k^{20} x^{k}}{(2 k+1) !}$$

Step-by-Step Solution

Answer

The radius of convergence is infinity (R = ∞). 2. What is the interval of convergence of the power series? The interval of convergence is all real numbers: $(-\infty, +\infty)$.

Step 1: Apply the Ratio Test

First, we apply the Ratio Test to find the radius of convergence. For the power series $$\sum \frac{k^{20} x^{k}}{(2 k+1) !}$$ we compute the limit, $$L = \lim_{k \to \infty} \left|\frac{a_{k+1}}{a_k}\right| $$ We have $$a_k = \frac{k^{20} x^{k}}{(2 k+1) !}$$ and $$a_{k+1} = \frac{(k+1)^{20} x^{k+1}}{(2 (k+1)+1) !}$$ Now, find the limit of the ratio as k approaches infinity: $$L = \lim_{k \to \infty} \left|\frac{\frac{(k+1)^{20} x^{k+1}}{(2 (k+1)+1) !}}{\frac{k^{20} x^{k}}{(2 k+1) !}}\right|$$

Step 2: Simplify the Limit Expression

We need to simplify the limit expression. We can do this by eliminating common factors and simplifying the expressions inside the limit: $$L = \lim_{k \to \infty} \left| \frac{(k+1)^{20} x^{k+1}(2 k+1) !}{k^{20} x^{k}(2 (k+1)+1) !}\right|$$ $$L = \lim_{k \to \infty} \left| x \frac{(k+1)^{20}}{k^{20}} \cdot \frac{(2 k+1) !}{(2 (k+1)+1) !} \right|$$

Step 3: Calculate the Limit

Now, calculate the limit as k approaches infinity: $$L = |x| \lim_{k \to \infty} \left(\frac{(k+1)^{20}}{k^{20}}\right) \lim_{k \to \infty} \left( \frac{(2 k+1) !}{(2 (k+1)+1) !}\right)$$ Observe that the second limit is $$\lim_{k \to \infty} \frac{(2 k+1) !}{(2 (k+1)+1) !} = \lim_{k \to \infty} \frac{1}{(2k+3)(2k+2)} = 0$$ Thus, the limit L converges to $$L= |x| (1) (0) = 0$$ According to the Ratio Test, the series converges when L < 1. Since L is always 0, the series always converges, regardless of the value of x. Therefore, the radius of convergence is infinity (R = ∞).

Step 4: Test the Endpoints

Since the radius of convergence is infinity, the power series converges for all x. Therefore, we do not need to test the endpoints. The interval of convergence is all real numbers: $$(-\infty, +\infty)$$