Mathematics

When you differentiate a power series term by term, you're applying the derivative to each term $a_n(x-5)^n$, which brings down the power and reduces the exponent by one. The key insight is that this operation doesn't change the underlying growth rate of the coefficients—you're just multiplying by $n$, which grows much slower than the exponential behavior that determines convergence. Therefore, the radius of convergence remains exactly $R$.

When you differentiate a power series term by term, you're applying the derivative to each term $a_n(x-5)^n$, which brings down the power and reduces the exponent by one. The key insight is that this operation doesn't change the underlying growth rate of the coefficients—you're just multiplying by $n$, which grows much slower than the exponential behavior that determines convergence. Therefore, the radius of convergence remains exactly $R$.