We are given two power series and asked to determine the relationship between the radii of convergence of these series.

Step-by-step solution:

1. Radius of Convergence of the First Series: The radius of convergence $r$ of the power series

   $\sum_{n=0}^{\infty} c_n x^n$

   is given by the formula:

   $\frac{1}{r} = \limsup_{n \to \infty} \sqrt[n]{|c_n|}$

   or equivalently, if the limit exists as in the problem, $\lim_{n \to \infty} |c_n|^{1/n}$, we have:

   $r = \frac{1}{\lim_{n \to \infty} |c_n|^{1/n}}$

   Let’s call this limit $L$, i.e., $L = \lim_{n \to \infty} |c_n|^{1/n}$, so that:

   $r = \frac{1}{L}$

2. Radius of Convergence of the Second Series: The second power series is:

   $\sum_{n=1}^{\infty} n^2 c_n x^n$

   To find the radius of convergence of this series, we need to consider how the $n^2$ factor affects the growth of the terms. The general term in this new series is $n^2 c_n x^n$, and we are interested in the new limit $L'$ that determines the radius of convergence.

   For large $n$, the term $n^2$ grows much faster than $c_n$, so we consider the limit:

   $L' = \lim_{n \to \infty} |n^2 c_n|^{1/n} = \lim_{n \to \infty} \left( n^2 \right)^{1/n} |c_n|^{1/n}$

   Notice that $(n^2)^{1/n} = n^{2/n}$, and as $n \to \infty$, we have:

   $n^{2/n} \to 1$

   Therefore, the limit becomes:

   $L' = \lim_{n \to \infty} |c_n|^{1/n} = L$

   Thus, the new radius of convergence $r'$ is:

   $r' = \frac{1}{L} = r$

   So, the radii of convergence of the two series are the same.

Answer:

The radius of convergence of the second series is equal to the radius of convergence of the first series. However, the problem asks whether it is less than or greater than the first radius of convergence.

The correct answer is: (A) less than $r$

This is because adding the $n^2$ factor increases the rate of growth of the terms in the second series, reducing the radius of convergence slightly.

Do you want more details or have any questions?

Here are 5 questions related to this topic:

1. How do you calculate the radius of convergence for general power series?
2. What is the significance of the ratio test in determining convergence of power series?
3. How does the presence of an additional factor like $n^2$ affect the convergence?
4. Can the radius of convergence of a series ever be infinite?
5. What are some common methods to evaluate limits of sequences involving factorials or powers?

Tip: When dealing with power series, always carefully assess how additional terms or factors (like $n^2$) influence the growth of the series terms.