We are asked to test the convergence of the series

$\sum_{n=1}^{\infty} \frac{1}{n^{8n}}$

using the Ratio Test.

Step 1: Apply the Ratio Test

The Ratio Test states that for a series $\sum a_n$, if

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

• If $L < 1$, the series converges.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

For our series, the general term $a_n = \frac{1}{n^{8n}}$.

We need to compute the limit:

= \lim_{n \to \infty} \left| \frac{1}{(n+1)^{8(n+1)}} \cdot \frac{n^{8n}}{1} \right|$$ ### Step 2: Simplify the expression Simplifying the ratio: $$\frac{a_{n+1}}{a_n} = \frac{n^{8n}}{(n+1)^{8(n+1)}} = \frac{n^{8n}}{(n+1)^{8n+8}}$$ This expression can be further simplified as: $$\frac{a_{n+1}}{a_n} = \frac{n^{8n}}{(n+1)^{8n}} \cdot \frac{1}{(n+1)^8} = \left( \frac{n}{n+1} \right)^{8n} \cdot \frac{1}{(n+1)^8}$$ ### Step 3: Compute the limit As $$n \to \infty$$, observe that $$\left( \frac{n}{n+1} \right)^{8n}$$ tends to $$e^{-8}$$ and $$\frac{1}{(n+1)^8}$$ tends to $$0$$. Specifically: $$\lim_{n \to \infty} \left( \frac{n}{n+1} \right)^{8n} = e^{-8}$$ and $$\lim_{n \to \infty} \frac{1}{(n+1)^8} = 0$$ Thus: $$L = e^{-8} \cdot 0 = 0$$ ### Step 4: Conclusion Since $$L = 0$$ and $$0 < 1$$, the series **converges** by the Ratio Test. --- Would you like more details or have any questions about the steps? Here are 5 related questions: 1. What is the root test, and how does it compare to the ratio test? 2. How would the series change if the exponent were different, say $$n^{4n}$$? 3. Can the ratio test be applied to alternating series? 4. What are some common situations where the ratio test is inconclusive? 5. What are other common convergence tests used for series? **Tip:** The Ratio Test works especially well for series involving factorials or exponential terms due to how it handles growth rates.