Math Problem Statement
Solve this problem
Solution
To determine the radius of convergence for the series
we can use the Root Test or Ratio Test to find the radius of convergence . The Root Test is particularly effective in this case because it can handle powers of more easily when the exponent grows as a multiple of .
The general term of the series is:
Applying the Root Test
The Root Test states that for a series , the radius of convergence can be found as:
For our term , we evaluate: [ \sqrt[n]{|a_n|} = \sqrt[n]{\frac{|x|^{4n}}{n^4}} = \frac{|x|^4}{n^{4/n}}. ]
As , , so:
For convergence, we need:
This implies:
Therefore, the radius of convergence is 1.
Conclusion
The radius of convergence of the series is:
Would you like a further breakdown or have any additional questions?
Here are five related questions that could expand your understanding:
- How would the convergence radius change if the exponent of were instead of ?
- What does the Root Test tell us about convergence at the boundary ?
- How does the Ratio Test apply to this series, and does it yield the same result?
- What is the significance of the alternating sign in terms of convergence behavior?
- How would the radius of convergence change if the denominator were instead of ?
Tip: When faced with series involving terms like , consider using the Root Test, as it often simplifies the analysis.
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Math Problem Analysis
Mathematical Concepts
Series
Convergence
Radius of Convergence
Formulas
Root Test formula: \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \)
Condition for convergence: \( |x| < 1 \)
Theorems
Root Test
Suitable Grade Level
Undergraduate Mathematics
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