The question is about the Region of Convergence (ROC) of the summation of two power series. Let's analyze the problem step by step.

1. There are two power series given:

   • $\sum a_n x^n$ with ROC $R_1$,
   • $\sum b_n x^n$ with ROC $R_2$, where $R_2 \neq R_1$.

2. We are tasked to find the ROC of the power series $\sum (a_n + b_n)x^n$.

Key Concept:

The ROC of a power series depends on the minimum radius of convergence of the individual series because the series must converge for both $\sum a_n x^n$ and $\sum b_n x^n$.

Thus, the ROC of $\sum (a_n + b_n)x^n$ will be: $\text{ROC} = \min \{R_1, R_2\}.$

Answer:

The correct option is: (b) min {R_1, R_2}

Do you want me to explain the reasoning behind the ROC concept or clarify any related concepts? Here are five questions that might help expand your understanding:

1. What is the definition of a power series and its radius of convergence?
2. Why is the ROC determined by the series with the smaller radius of convergence?
3. How does the radius of convergence change if two series are multiplied instead of added?
4. Can the ROC of a power series be infinite? Under what conditions?
5. What happens to the ROC if we scale the series coefficients by a constant factor?

Tip: When working with power series, always consider the individual convergence of each term involved in the combination.