Prove the radius of convergence of a specific power series is exactly 1 through a detailed mathematical proof by contradiction | Step-by-Step Solution

What We're Solving:

We need to prove that a specific power series has a radius of convergence of exactly 1, using a proof by contradiction approach.

The Approach:

This is a proof-writing assignment using proof by contradiction, which works beautifully for radius of convergence problems:

• The Logic: We assume the opposite of what we want to prove, then show this leads to an impossible situation
• For radius problems: We'll assume R ≠ 1 (meaning R > 1 OR R < 1), then show each case creates a contradiction

Step-by-Step Framework:

1. Set Up Your Proof Structure

• Opening: State what you're proving clearly
• Method declaration: "We proceed by contradiction..."
• Assumption: "Suppose R ≠ 1"

2. Break Into Cases

Your proof should handle:

• Case 1: Assume R > 1
• Case 2: Assume R < 1

3. For Each Case, Use Key Tools:

• Ratio Test or Root Test - These directly relate to radius of convergence
• Properties of your specific coefficient sequence - What makes your series special?
• Convergence/divergence at boundary points - Test x = 1 and x = -1

4. Show the Contradiction

For each case, demonstrate that your assumption leads to something impossible, such as:

• A convergent series that should diverge
• A coefficient pattern that violates known properties
• A violation of a convergence test

5. Strong Conclusion Structure

• "Since both cases lead to contradictions..."
• "Therefore, our assumption was false..."
• "Hence, R = 1 exactly."

The Framework (Your Outline):

I. Introduction & Setup

• State the power series explicitly
• Define what "radius of convergence equals 1" means
• Announce proof by contradiction

II. Case 1: R > 1

• What this assumption implies about convergence
• Apply appropriate convergence tests
• Show why this creates a contradiction

III. Case 2: R < 1

• What this assumption implies
• Use series properties or tests
• Demonstrate the resulting impossibility

IV. Conclusion

• Synthesize why both cases fail
• State that R = 1 is the only possibility

Memory Tip:

Think of proof by contradiction like being a detective! You're saying "If the radius ISN'T 1, then something fishy should happen." When you find that "fishy thing" (the contradiction), you've solved the case - the radius MUST be 1!

Pro tip: The specific coefficient sequence in your series holds the key to finding those contradictions. Look for patterns, growth rates, or special properties that make certain convergence behaviors impossible!