Question

What is the radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)?

• A. \(0\)
• B. \(1\)
• C. \(\infty\)
• D. \(e\)

Hint:

\(\sum \frac{x^n}{n!} = e^x\)
Exponential series converges for all real numbers \(x\)

Answer 1

The Correct Option is C

Concept: Radius of convergence using Ratio Test
For a power series: \[ \sum a_n x^n \] the radius of convergence \(R\) is found using: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
Step 1: Identify coefficients
\[ a_n = \frac{1}{n!} \]
Step 2: Apply Ratio Test
\[ \left| \frac{a_{n+1}}{a_n} \right} = \frac{1}{(n+1)!} \cdot n! = \frac{1}{n+1} \]
Step 3: Take the limit
\[ \lim_{n \to \infty} \frac{1}{n+1} = 0 \]
Step 4: Interpretation
Since the limit is 0 for all \(x\), the series converges for all real \(x\). Conclusion: \[ R = \infty \]