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.