Math Problem Statement

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The radius of convergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^n x^{4n}}{n^4} \) is

Solution

To determine the radius of convergence for the series

n=1(1)nx4nn4,\sum_{n=1}^{\infty} \frac{(-1)^n x^{4n}}{n^4},

we can use the Root Test or Ratio Test to find the radius of convergence RR. The Root Test is particularly effective in this case because it can handle powers of xx more easily when the exponent grows as a multiple of nn.

The general term of the series is: an=(1)nx4nn4.a_n = \frac{(-1)^n x^{4n}}{n^4}.

Applying the Root Test

The Root Test states that for a series an\sum a_n, the radius of convergence RR can be found as: 1R=limnann.\frac{1}{R} = \lim_{n \to \infty} \sqrt[n]{|a_n|}.

For our term an=x4nn4a_n = \frac{x^{4n}}{n^4}, we evaluate: [ \sqrt[n]{|a_n|} = \sqrt[n]{\frac{|x|^{4n}}{n^4}} = \frac{|x|^4}{n^{4/n}}. ]

As nn \to \infty, n4/n1n^{4/n} \to 1, so: limnann=x4.\lim_{n \to \infty} \sqrt[n]{|a_n|} = |x|^4.

For convergence, we need: x4<1.|x|^4 < 1.

This implies: x<1.|x| < 1.

Therefore, the radius of convergence RR is 1.

Conclusion

The radius of convergence of the series is: 1.\boxed{1}.

Would you like a further breakdown or have any additional questions?

Here are five related questions that could expand your understanding:

  1. How would the convergence radius change if the exponent of xx were 2n2n instead of 4n4n?
  2. What does the Root Test tell us about convergence at the boundary x=R|x| = R?
  3. How does the Ratio Test apply to this series, and does it yield the same result?
  4. What is the significance of the alternating sign (1)n(-1)^n in terms of convergence behavior?
  5. How would the radius of convergence change if the denominator were n2n^2 instead of n4n^4?

Tip: When faced with series involving terms like xknx^{kn}, 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