To determine the radius and interval of convergence of the power series given by:

$\sum \frac{x^{2k + 1}}{2^{k - 1}}$

we can approach this problem by finding the radius of convergence $R$ using the Ratio Test.

Step-by-Step Solution

1. Rewrite the General Term of the Series: The general term $a_k$ of the series is: $a_k = \frac{x^{2k + 1}}{2^{k - 1}}$

2. Apply the Ratio Test: The Ratio Test states that the radius of convergence $R$ can be found using: $\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \frac{1}{R}$ To proceed, compute $a_{k+1}$: $a_{k+1} = \frac{x^{2(k+1) + 1}}{2^{(k+1) - 1}} = \frac{x^{2k + 3}}{2^k}$

3. Calculate the Ratio $\frac{a_{k+1}}{a_k}$:

   = \frac{x^2 \cdot 2^{k - 1}}{2^k} = x^2 \cdot \frac{1}{2} = \frac{x^2}{2}$$

4. Find the Radius of Convergence $R$: For the series to converge, the limit must satisfy: $\left| \frac{x^2}{2} \right| < 1$ Simplifying, we find: $|x^2| < 2 \Rightarrow |x| < \sqrt{2}$ Therefore, the radius of convergence $R$ is: $R = \sqrt{2}$

So, the radius of convergence $R$ is: $R = \sqrt{2}$

Interval of Convergence

To determine the interval of convergence, consider values within $|x| < \sqrt{2}$. We would need further testing at $x = \pm \sqrt{2}$ to determine convergence at these endpoints, but based on the radius, we can conclude the open interval as $(-\sqrt{2}, \sqrt{2})$.

Would you like additional details on the interval test at the endpoints or have any other questions?

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Related Questions:

1. How does the Ratio Test determine the convergence radius?
2. What alternative methods can be used to find the radius of convergence?
3. What role do endpoints play in convergence intervals for power series?
4. How does the exponent form $2k + 1$ affect convergence in this power series?
5. Why is it essential to check endpoints separately when determining interval convergence?

Tip: For power series, always test convergence at endpoints separately after finding the radius to ensure the full interval is accurately determined.