To find the interval of convergence and radius of convergence for the series

$\sum_{n=1}^\infty \frac{(3x-1)^n}{n^2},$

we proceed as follows:

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Step 1: Analyze the given series

The series can be rewritten as:

$\sum_{n=1}^\infty \frac{(3x-1)^n}{n^2}.$

This is a power series in $(3x-1)$. The ratio test or root test can be used to determine the convergence.

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Step 2: Apply the root test

The root test examines the limit:

$\limsup_{n \to \infty} \sqrt[n]{|a_n|},$

where $a_n = \frac{(3x-1)^n}{n^2}$.

First, extract the general term: $|a_n| = \frac{|3x-1|^n}{n^2}.$

Now compute the nth root: [ \sqrt[n]{|a_n|} = \sqrt[n]{\frac{|3x-1|^n}{n^2}} = \frac{|3x-1|}{n^{2/n}}. ]

As $n \to \infty$, $n^{2/n} \to 1$, so: $\limsup_{n \to \infty} \sqrt[n]{|a_n|} = |3x-1|.$

For convergence of the series, the root test requires: $|3x-1| < 1.$

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Step 3: Solve for $x$ in terms of the inequality

The inequality $|3x-1| < 1$ can be rewritten as: $-1 < 3x - 1 < 1.$

Add 1 to all parts: $0 < 3x < 2.$

Divide by 3: $0 < x < \frac{2}{3}.$

Thus, the radius of convergence is: $R = \frac{1}{3}.$

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Step 4: Test endpoints

To determine the interval of convergence, we test the endpoints $x = 0$ and $x = \frac{2}{3}$:

1. At $x = 0$: Substituting $x = 0$ gives $(3x-1)^n = (-1)^n$: $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}.$ This series converges by the alternating series test and the p-series test ($p = 2 > 1$).

2. At $x = \frac{2}{3}$: Substituting $x = \frac{2}{3}$ gives $(3x-1)^n = 1^n = 1$: $\sum_{n=1}^\infty \frac{1}{n^2}.$ This series converges by the p-series test ($p = 2 > 1$).

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Final Answer:

• Interval of Convergence: $[0, \frac{2}{3}]$.
• Radius of Convergence: $R = \frac{1}{3}$.

Would you like additional details about any step? Here are some related questions to explore:

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5 Related Questions:

1. How does the root test differ from the ratio test in determining convergence?
2. Why do we need to test endpoints separately when finding the interval of convergence?
3. What is the significance of the p-series test, and when can it be applied?
4. How does absolute convergence affect the interval of convergence?
5. Can a series converge at one endpoint but not the other? Provide examples.

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Tip:

Always verify endpoints separately when finding the interval of convergence; convergence behavior at endpoints can differ significantly.