1. The problem is to determine the radius of convergence of a power series and find the value of the series, its derivative, and its integral at each point within that radius. 2. The radius of convergence $R$ of a power series $\sum a_n (x - c)^n$ can be found using the formula $$R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}.$$ This radius tells us the interval $|x - c| < R$ where the series converges. 3. Once $R$ is found, the original series converges for $|x - c| < R$. The differentiated series is $$\sum n a_n (x - c)^{n-1}$$ and the integrated series is $$C + \sum \frac{a_n}{n+1} (x - c)^{n+1}$$ where $C$ is the constant of integration. 4. The radius of convergence for the differentiated and integrated series is the same as the original series. 5. To find the value of the series at each point within the radius, substitute $x$ into the series and sum the terms. 6. Similarly, substitute $x$ into the differentiated and integrated series to find their values. 7. Important: The series converges absolutely and uniformly on any closed interval inside $|x - c| < R$. Since the problem does not specify the exact series, this is the general method to solve it.