Math Problem Statement

Find all the values of x such that the given series would converge: ∑_{n=1}^{∞} (4x)^n / n^8.

Solution

The given series is: ∑n=1∞(4x)nn8\sum_{n=1}^{\infty} \frac{(4x)^n}{n^8}

We need to find the values of xx for which this series converges. To determine the convergence of this series, we'll use the Ratio Test. The Ratio Test states that a series ∑an\sum a_n converges if: lim⁡n→∞∣an+1an∣<1\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1

Let's define the general term of the series: an=(4x)nn8a_n = \frac{(4x)^n}{n^8}

Applying the Ratio Test

Calculate the ratio an+1an\frac{a_{n+1}}{a_n}:

= \frac{(4x)^{n+1} \cdot n^8}{(4x)^n \cdot (n+1)^8} = (4x) \cdot \frac{n^8}{(n+1)^8}$$ Taking the limit as $$n$$ approaches infinity: $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |4x| \cdot \lim_{n \to \infty} \frac{n^8}{(n+1)^8}$$ Since $$\lim_{n \to \infty} \frac{n^8}{(n+1)^8} = 1$$, we have: $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |4x|$$ For convergence using the Ratio Test: $$|4x| < 1$$ $$-1 < 4x < 1$$ $$-\frac{1}{4} < x < \frac{1}{4}$$ ### Endpoints Check Now we check the convergence at the endpoints $$x = -\frac{1}{4}$$ and $$x = \frac{1}{4}$$. 1. **At $$x = -\frac{1}{4}$$:** The series becomes: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^8}$$ This series converges by the Alternating Series Test because $$\frac{1}{n^8}$$ decreases monotonically to zero. 2. **At $$x = \frac{1}{4}$$:** The series becomes: $$\sum_{n=1}^{\infty} \frac{1}{n^8}$$ This series converges by the $$p$$-series test with $$p = 8 > 1$$. ### Conclusion The interval of convergence is: - From $$x = -\frac{1}{4}$$, left endpoint included (Y). - To $$x = \frac{1}{4}$$, right endpoint included (Y). So, the answers are: - From $$x = -\frac{1}{4}$$, left end included: Y - To $$x = \frac{1}{4}$$, right end included: Y Would you like more details on this process or have any questions? Here are some related questions to consider: 1. What is the Alternating Series Test and how does it work? 2. How do we apply the Ratio Test to determine convergence? 3. What is the significance of a $$p$$-series in convergence tests? 4. Why does the series converge at the endpoints in this case? 5. Can the Root Test be used as an alternative to the Ratio Test for this series? **Tip:** When checking endpoint convergence, it's essential to consider different tests like the Alternating Series Test or $$p$$-series test to confirm the behavior at those points.

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Math Problem Analysis

Mathematical Concepts

Series Convergence
Ratio Test
Alternating Series Test
p-series

Formulas

Ratio Test: lim(n→∞) |a_(n+1) / a_n| < 1
Alternating Series Test
p-series test: convergence if p > 1

Theorems

Ratio Test
Alternating Series Test
p-series Test

Suitable Grade Level

Undergraduate Mathematics

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