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Chapter 10: Problem 27

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

Short Answer

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

Step by step solution

01

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: $$L = \lim_{k \to \infty} \left|\frac{\frac{(k+1)^{20} x^{k+1}}{(2 (k+1)+1) !}}{\frac{k^{20} x^{k}}{(2 k+1) !}}\right|$$
02

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: $$L = \lim_{k \to \infty} \left| \frac{(k+1)^{20} x^{k+1}(2 k+1) !}{k^{20} x^{k}(2 (k+1)+1) !}\right|$$ $$L = \lim_{k \to \infty} \left| x \frac{(k+1)^{20}}{k^{20}} \cdot \frac{(2 k+1) !}{(2 (k+1)+1) !} \right|$$
03

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 = 鈭�).
04

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: $$(-\infty, +\infty)$$

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Most popular questions from this chapter

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