To solve this problem, let's analyze the given series and determine both the interval and radius of convergence.

Given Series:

$\sum_{n=1}^{\infty} \frac{(-1)^n n^7 x^n}{\sqrt{n + 9}}$

To find the interval and radius of convergence, we will use the Ratio Test. The Ratio Test states that for a series $\sum a_n$, the series converges if: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$ and diverges if this limit is greater than 1.

Step 1: Define $a_n$

In this series, $a_n = \frac{(-1)^n n^7 x^n}{\sqrt{n + 9}}$

Step 2: Apply the Ratio Test

We need to compute: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

Calculating $a_{n+1}$: $a_{n+1} = \frac{(-1)^{n+1} (n+1)^7 x^{n+1}}{\sqrt{(n+1) + 9}}$

Now, taking the ratio $\left| \frac{a_{n+1}}{a_n} \right|$: $\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(-1)^{n+1} (n+1)^7 x^{n+1} / \sqrt{n+10}}{(-1)^n n^7 x^n / \sqrt{n+9}} \right|$ $= \left| \frac{(n+1)^7 x^{n+1} \sqrt{n+9}}{n^7 x^n \sqrt{n+10}} \right|$ $= |x| \cdot \frac{(n+1)^7}{n^7} \cdot \sqrt{\frac{n+9}{n+10}}$

Step 3: Simplify the Limit

As $n \to \infty$, $\frac{(n+1)^7}{n^7} \approx 1$ and $\sqrt{\frac{n+9}{n+10}} \approx 1$. Thus, the limit simplifies to: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |x|$

Step 4: Find the Radius of Convergence

For convergence, we need: $|x| < 1$ Thus, the radius of convergence $R$ is 1.

Step 5: Determine the Interval of Convergence

The radius of convergence suggests that $-1 < x < 1$.

To determine the interval of convergence, we check the endpoints $x = -1$ and $x = 1$:

1. At $x = 1$: $\sum_{n=1}^{\infty} \frac{(-1)^n n^7}{\sqrt{n+9}}$ This is an alternating series with terms that do not approach zero as $n \to \infty$, so it diverges at $x = 1$.

2. At $x = -1$: $\sum_{n=1}^{\infty} \frac{(-1)^n n^7 (-1)^n}{\sqrt{n+9}} = \sum_{n=1}^{\infty} \frac{n^7}{\sqrt{n+9}}$ This series also diverges because the terms $\frac{n^7}{\sqrt{n+9}}$ do not approach zero as $n \to \infty$.

Thus, the interval of convergence is: $(-1, 1)$

Final Answer:

• Interval of Convergence: $x = -1$ to $x = 1$
• Radius of Convergence: $R = 1$

Would you like further details on any step?

Related Questions:

1. How does the Ratio Test help determine convergence for power series?
2. Why do we need to check endpoints separately when determining intervals of convergence?
3. What would happen if the series involved a term like $x^{2n}$ instead of $x^n$?
4. How do we apply similar methods to find convergence intervals for other series types, like Taylor or Fourier series?
5. What are some common mistakes when using the Ratio Test for convergence?

Tip:

Always verify convergence at endpoints separately, as the Ratio Test only gives information about the open interval of convergence.