Math Problem Statement
Solution
To determine the largest open interval of convergence and the radius of convergence for the given series:
we follow these steps:
Step 1: Ratio Test for Convergence
The ratio test is used to analyze the convergence of the series.
Let . Compute the ratio of consecutive terms:
\lim_{n \to \infty} \left| \frac{\frac{(3x)^{n+1}}{(n+1)!}}{\frac{(3x)^n}{n!}} \right| = \lim_{n \to \infty} \left| \frac{(3x)^{n+1} \cdot n!}{(3x)^n \cdot (n+1)!} \right|$$ Simplify: $$= \lim_{n \to \infty} \left| \frac{3x \cdot (3x)^n}{(n+1) \cdot (3x)^n} \right| = \lim_{n \to \infty} \left| \frac{3x}{n+1} \right|$$ $$= 0 \quad \text{for all } x.$$ Since this limit is $$0$$, which is less than $$1$$, the series converges for all values of $$x$$. --- ### Step 2: Radius of Convergence The radius of convergence is infinite because the ratio test shows the series converges for all $$x$$. Thus: $$R = \infty.$$ --- ### Step 3: Interval of Convergence Since the radius of convergence is infinite, the series converges for all $$x \in \mathbb{R}$$. The largest open interval of convergence is: $$(-\infty, \infty).$$ --- ### Final Answer: - **Radius of Convergence**: $$R = \infty$$ - **Interval of Convergence**: $$(- \infty, \infty)$$ Let me know if you have any questions or would like detailed explanations of any step! Here are related questions for further exploration: 1. What is the behavior of this series when $$x = 0$$? 2. How does the factorial term $$n!$$ in the denominator affect convergence? 3. Can you explain why the ratio test guarantees absolute convergence? 4. What happens if the term $$(3x)^n$$ is replaced by $$x^n$$? 5. How does the exponential growth of the numerator compare to the factorial in the denominator? **Tip**: The factorial $$n!$$ grows faster than any polynomial or exponential of $$n$$, which often leads to rapid convergence for series involving $$n!$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Series Convergence
Radius of Convergence
Interval of Convergence
Ratio Test
Formulas
Ratio test formula: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)
Definition of radius of convergence \( R \)
Theorems
Ratio Test
Convergence of Exponential Series
Suitable Grade Level
Undergraduate Mathematics
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