Determine the radius of convergence of ∑ x^n / n!.

Determine the radius of convergence of ∑ x^n / n!.

• A. ∞
• B. 1
• C. 0
• D. 2

Answer: (D)

Think about how fast the terms shrink as n grows. For a power series ∑ c_n x^n, the radius of convergence comes from the ratio of successive terms. Here c_n = 1/n!. The ratio of successive terms is (x^{n+1}/(n+1)! ) ÷ (x^n/n!) = x/(n+1). As n → ∞, this ratio tends to 0 for any fixed x. The ratio test then tells us the series converges for every real (and complex) x because the limit is 0, which is certainly less than 1. So the radius of convergence is infinite, meaning the series defines an entire function. This aligns with the known expansion of e^x.

Think about how fast the terms shrink as n grows. For a power series ∑ c_n x^n, the radius of convergence comes from the ratio of successive terms.

Here c_n = 1/n!. The ratio of successive terms is

(x^{n+1}/(n+1)! ) ÷ (x^n/n!) = x/(n+1).

As n → ∞, this ratio tends to 0 for any fixed x. The ratio test then tells us the series converges for every real (and complex) x because the limit is 0, which is certainly less than 1.

So the radius of convergence is infinite, meaning the series defines an entire function. This aligns with the known expansion of e^x.