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The task in this exercise is to determine the **radius of convergence** of the given power series:

\[
P(x) = \sum_{k=0}^{\infty} \frac{k^2}{2k + 1} \cdot (x)^k
\]

To find the radius of convergence \( R \), we use the **root test** or the **ratio test**. In this case, the ratio test is particularly useful. The ratio test is based on the following formula for the radius of convergence:

\[
\frac{1}{R} = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|
\]

Where \( a_k = \frac{k^2}{2k + 1} \).

Let's apply the ratio test step-by-step to this series to find the radius of convergence.The radius of convergence \( R \) for the given power series is:

\[
R = 1
\]

This means the series converges for \( |x| < 1 \).

Would you like any further details on this solution or explanation?

Here are five questions to explore this topic further:
1. What is the significance of the radius of convergence in a power series?
2. How does the ratio test work for determining convergence?
3. Can we apply the root test to this series as an alternative method?
4. What happens to the series when \( x = \pm 1 \)?
5. How does the behavior of the series change outside the radius of convergence?

**Tip:** When using the ratio test, if the limit results in 1, it indicates that the series may converge conditionally at the boundary points \( x = \pm R \). Always check for convergence at the endpoints separately.

Ermitteln Sie den Konvergenzradius der Potenzreihe: \[ P(x) = \sum_{k=0}^{\infty} \frac{k^2}{2k + 1} \cdot (x)^k \]

This page provides a step-by-step guide to determining the radius of convergence for the power series \( P(x) = \sum_{k=0}^{\infty} \frac{k^2}{2k + 1} \cdot (x)^k \). It explores key mathematical concepts such as the Ratio Test and its application. Suitable for college-level mathematics students.

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