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The given image presents a series with multiple terms, and the series is noted to be convergent. Let's break it down:

The general structure of the series is:

\[
\left( \frac{2^2}{1^2} - \frac{2}{1} \right)^{-1} + \left( \frac{3^3}{2^3} - \frac{3}{2} \right)^{-2} + \left( \frac{4^4}{3^4} - \frac{4}{3} \right)^{-3} + \cdots
\]

We can simplify each term:

### First term:
\[
\left( \frac{2^2}{1^2} - \frac{2}{1} \right)^{-1} = \left( \frac{4}{1} - 2 \right)^{-1} = \left( 4 - 2 \right)^{-1} = 2^{-1} = \frac{1}{2}
\]

### Second term:
\[
\left( \frac{3^3}{2^3} - \frac{3}{2} \right)^{-2} = \left( \frac{27}{8} - \frac{3}{2} \right)^{-2}
\]

First, find a common denominator for \( \frac{27}{8} - \frac{3}{2} \):

\[
\frac{3}{2} = \frac{12}{8}, \quad \therefore \quad \frac{27}{8} - \frac{12}{8} = \frac{15}{8}
\]

Thus, the second term becomes:
\[
\left( \frac{15}{8} \right)^{-2} = \left( \frac{8}{15} \right)^2 = \frac{64}{225}
\]

### Third term:
\[
\left( \frac{4^4}{3^4} - \frac{4}{3} \right)^{-3} = \left( \frac{256}{81} - \frac{4}{3} \right)^{-3}
\]

Find a common denominator for \( \frac{256}{81} - \frac{4}{3} \):

\[
\frac{4}{3} = \frac{108}{81}, \quad \therefore \quad \frac{256}{81} - \frac{108}{81} = \frac{148}{81}
\]

Thus, the third term becomes:
\[
\left( \frac{148}{81} \right)^{-3} = \left( \frac{81}{148} \right)^3 = \frac{531441}{3241792}
\]

### General pattern and convergence:
The general form of the series appears to involve powers that increase with each term. Specifically, the denominators grow rapidly as powers of the base numbers increase. This suggests that the series is decaying quickly and thus is likely convergent, as indicated by the final answer.

### Conclusion:
The series converges because the terms decrease in magnitude rapidly.

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. What are the common tests for determining convergence of a series?
2. How does the ratio test apply to this series?
3. Could the series be rewritten in a simpler form for easier evaluation?
4. How does the growth of powers in the series contribute to its convergence?
5. What other examples of convergent series behave similarly?

**Tip**: When simplifying terms in a series, ensure all components are reduced properly before applying convergence tests.

( (2^2 / 1^2 - 2/1)^{-1} ) + ( (3^3 / 2^3 - 3/2)^{-2} ) + ( (4^4 / 3^4 - 4/3)^{-3} ) + ...

Learn how to simplify and analyze a complex series involving exponentiation and fraction subtraction. The series, which includes terms like (2^2/1^2 - 2/1)^{-1}, is shown to be convergent by using techniques like fraction simplification and series convergence tests. Suitable for advanced students.

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