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

$\sum_{n=1}^{\infty} \frac{(10x)^n}{n^5},$

we use the ratio test.

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1. Apply the Ratio Test

The ratio test examines the limit of consecutive terms: $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.$ Here, $a_n = \frac{(10x)^n}{n^5}$. So: $a_{n+1} = \frac{(10x)^{n+1}}{(n+1)^5}.$

The ratio becomes: $\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(10x)^{n+1}}{(n+1)^5} \cdot \frac{n^5}{(10x)^n} \right| = \left| \frac{10x \cdot n^5}{(n+1)^5} \right|.$

Simplify: $\left| \frac{a_{n+1}}{a_n} \right| = |10x| \cdot \frac{n^5}{(n+1)^5}.$

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2. Evaluate the Limit

As $n \to \infty$, $\frac{n^5}{(n+1)^5} \to 1.$

Thus: $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |10x| \cdot 1 = |10x|.$

For convergence, the ratio test requires: $L < 1 \implies |10x| < 1 \implies |x| < \frac{1}{10}.$

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3. Determine the Interval and Radius of Convergence

• Radius of Convergence: $R = \frac{1}{10}$.
• Interval of Convergence: $|x| < \frac{1}{10}$ implies $x \in \left(-\frac{1}{10}, \frac{1}{10}\right)$.

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4. Check the Endpoints

To find the exact interval, we need to check $x = \pm \frac{1}{10}$.

At $x = \frac{1}{10}$:

The series becomes: $\sum_{n=1}^\infty \frac{(10 \cdot \frac{1}{10})^n}{n^5} = \sum_{n=1}^\infty \frac{1}{n^5}.$ This is a $p$-series with $p = 5 > 1$, which converges.

At $x = -\frac{1}{10}$:

The series becomes: $\sum_{n=1}^\infty \frac{(10 \cdot -\frac{1}{10})^n}{n^5} = \sum_{n=1}^\infty \frac{(-1)^n}{n^5}.$ This is an alternating $p$-series with $p = 5 > 1$, which converges.

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Final Answer

• Interval of Convergence: $\left[ -\frac{1}{10}, \frac{1}{10} \right]$.
• Radius of Convergence: $R = \frac{1}{10}$.

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Do you want further details or have any questions? Here are five related questions:

1. How does the ratio test work for different series?
2. Can you explain the $p$-series test for convergence?
3. What happens if $p < 1$ in a $p$-series?
4. How do alternating series tests determine convergence?
5. How do we interpret the radius and interval of convergence practically?

Tip: Always check the endpoints separately when determining the interval of convergence, as they often behave differently than the interior points.