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To determine if the given series converges, let's analyze it.

The series is:

\[
\sum_{n=1}^{\infty} \frac{5^{n-1}}{3^{n+1}}
\]

### Step 1: Rewrite the General Term
We can rewrite the general term in a simpler form to see if it resembles a geometric series. Notice that:

\[
\frac{5^{n-1}}{3^{n+1}} = \frac{5^{n-1}}{3 \cdot 3^n} = \frac{1}{3} \cdot \frac{5^{n-1}}{3^n}
\]

Now we can rewrite \(\frac{5^{n-1}}{3^n}\) as \((\frac{5}{3})^{n-1}\). Thus, the term becomes:

\[
\frac{1}{3} \cdot \left(\frac{5}{3}\right)^{n-1}
\]

### Step 2: Recognize the Geometric Series Form
The series now looks like:

\[
\sum_{n=1}^{\infty} \frac{1}{3} \left(\frac{5}{3}\right)^{n-1}
\]

This is a geometric series with:
- First term \( a = \frac{1}{3} \)
- Common ratio \( r = \frac{5}{3} \)

### Step 3: Check for Convergence
For a geometric series \(\sum_{n=0}^{\infty} ar^n\) to converge, the common ratio \( |r| \) must be less than 1. In this case:

\[
\left|\frac{5}{3}\right| > 1
\]

Since \(\frac{5}{3}\) is greater than 1, the series diverges.

### Conclusion
The series \(\sum_{n=1}^{\infty} \frac{5^{n-1}}{3^{n+1}}\) **diverges** because the common ratio \(r = \frac{5}{3}\) is greater than 1.

Would you like further details on this process or have any questions?

Here are some related questions to explore further:
1. What conditions are necessary for a geometric series to converge?
2. How does the ratio test apply to series with similar terms?
3. What methods exist for determining convergence of non-geometric series?
4. How can we simplify complex fractions in series terms?
5. How does the convergence of a series change if we alter the base of exponents?

**Tip:** Always check if a series has a common ratio and whether it’s less than 1 when dealing with geometric series to determine convergence.

Determine if the series \( \sum_{n=1}^{\infty} \frac{5^{n-1}}{3^{n+1}} \) converges, and if so to which values.

Determine whether the series \( \sum_{n=1}^{\infty} \frac{5^{n-1}}{3^{n+1}} \) converges. This problem involves concepts of geometric series and convergence criteria. Detailed steps to check the convergence of this infinite series are provided, using the common ratio condition.

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