determine whether converges or diverges (2/n)(1/n - 3)(1/n -1)^-2 + (2/n)^-1 - (1)^-1

To determine whether the given series or expression converges or diverges, let's first analyze and simplify the expression:

Given expression:

\[
\frac{2}{n} \left(\frac{1}{n} - 3\right) \left(\frac{1}{n} - 1\right)^{-2} + \left(\frac{2}{n}\right)^{-1} - 1^{-1}
\]

### Step 1: Simplify each term

1. **Simplify the first term**:
   \[
   \frac{2}{n} \left(\frac{1}{n} - 3\right) \left(\frac{1}{n} - 1\right)^{-2}
   \]

   Let's rewrite the expression inside the parentheses:
   \[
   \frac{1}{n} - 3 \quad \text{and} \quad \left(\frac{1}{n} - 1\right)^{-2}
   \]
   As \(n\) becomes large, \(\frac{1}{n}\) approaches 0, making \(\frac{1}{n} - 3\) approximately \(-3\) and \(\frac{1}{n} - 1\) approximately \(-1\). Hence, the expression simplifies to:
   \[
   \approx \frac{2}{n} \times (-3) \times (-1)^{-2} = \frac{6}{n} \times 1 = \frac{6}{n}
   \]

2. **Simplify the second term**:
   \[
   \left(\frac{2}{n}\right)^{-1} = \frac{n}{2}
   \]

3. **Simplify the third term**:
   \[
   1^{-1} = 1
   \]

### Step 2: Combine the simplified terms

Putting everything together:
\[
\frac{6}{n} + \frac{n}{2} - 1
\]

### Step 3: Analyze the behavior as \(n\) approaches infinity

- The term \(\frac{6}{n}\) approaches 0 as \(n\) becomes large.
- The term \(\frac{n}{2}\) grows without bound as \(n\) increases.
- The constant term \(-1\) remains the same.

### Conclusion:
The dominant term in the expression is \(\frac{n}{2}\), which grows without bound as \(n\) increases. Therefore, the overall expression diverges as \(n\) approaches infinity.

The given expression **diverges**.

Would you like more details or have any questions? 

Here are 5 related questions to further explore the topic:
1. How do you determine if a series converges or diverges?
2. What is the difference between absolute and conditional convergence?
3. Can a series diverge if its individual terms go to zero?
4. How does the comparison test help in determining the convergence of a series?
5. What role does the ratio test play in series convergence?

**Tip:** Always check the behavior of the leading terms when determining the convergence of an expression as \(n\) approaches infinity.

Learn how to determine whether a series converges or diverges with a detailed mathematical analysis. This problem involves analyzing the behavior of series terms and applying limit definitions. Suitable for undergraduate-level students and anyone interested in advanced mathematical analysis.

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