To determine the convergence or divergence of the series

$\sum_{n=1}^{\infty} \frac{1}{5^n}$

using the Ratio Test, let's analyze the setup:

1. First, identify the general term $a_n = \frac{1}{5^n}$.

2. The Ratio Test states that for a series with terms $a_n$, we calculate the limit

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

   If $L < 1$, the series converges absolutely. If $L > 1$ or $L = \infty$, the series diverges. If $L = 1$, the test is inconclusive.

Step 1: Set up the ratio $\frac{a_{n+1}}{a_n}$

For our series: $a_n = \frac{1}{5^n}$ and $a_{n+1} = \frac{1}{5^{n+1}}$

Step 2: Calculate $\frac{a_{n+1}}{a_n}$

$\frac{a_{n+1}}{a_n} = \frac{\frac{1}{5^{n+1}}}{\frac{1}{5^n}} = \frac{1}{5}$

Step 3: Take the limit

Since $\frac{a_{n+1}}{a_n} = \frac{1}{5}$ for all $n$, we get

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{5}$

Conclusion

Since $L = \frac{1}{5} < 1$, the Ratio Test tells us that the series converges.

Would you like more details or have questions about any steps?

Here are some related questions to deepen understanding:

1. How does the Ratio Test compare to other convergence tests, like the Root Test?
2. What would happen if the base of the exponent were a different number, like 10, instead of 5?
3. Can we conclude absolute convergence for all values of $L < 1$, or are there exceptions?
4. How can we apply the Ratio Test to series with factorial terms, such as $\sum \frac{n!}{n^n}$?
5. Why is the Ratio Test inconclusive when $L = 1$?

Tip: Always remember that the Ratio Test is particularly useful for series involving exponential and factorial terms, where ratios simplify the analysis effectively.