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The given equation is:

\[
C = 60 \cdot \sqrt{\frac{1 + x + x^2 + x^3 + \cdots + x^{60}}{1 + x^{-1} + x^{-2} + x^{-3} + \cdots + x^{-60}}}, \quad (x \in \mathbb{R}^+)
\]

### Explanation:

1. The equation consists of a sum in both the numerator and the denominator, running from powers of \(x\) and \(x^{-1}\), respectively.
   
2. The numerator is a finite geometric series with the terms from \(x^0\) to \(x^{60}\), i.e., \(1 + x + x^2 + \cdots + x^{60}\).

3. The denominator is also a geometric series, but it consists of negative powers of \(x\), i.e., \(1 + x^{-1} + x^{-2} + \cdots + x^{-60}\).

4. The equation is multiplied by a factor of 60 and involves a square root across the entire fraction.

### Simplification Approach:

- **Numerator**: This is a geometric series with first term \(1\) and common ratio \(x\), running up to \(x^{60}\). The sum of a geometric series \(S_n = a + ar + ar^2 + \dots + ar^n\) is given by:

\[
S = \frac{1 - r^{n+1}}{1 - r} \quad \text{for} \quad r \neq 1
\]

Thus, the sum of the numerator is:

\[
S_{\text{numerator}} = \frac{1 - x^{61}}{1 - x}, \quad \text{for} \, x \neq 1
\]

- **Denominator**: This is also a geometric series but with terms involving negative powers of \(x\). The sum of this geometric series is:

\[
S_{\text{denominator}} = \frac{1 - x^{-61}}{1 - x^{-1}}
\]

Using \(x^{-1} = \frac{1}{x}\), the denominator can be rewritten and simplified.

Would you like me to fully simplify this expression, or would you like an explanation of any particular part of this equation? 

### Questions:
1. What is the sum of a general geometric series?
2. How does the equation change if \(x = 1\)?
3. How can you rewrite the sum of negative powers of \(x\)?
4. What are the conditions for convergence of a geometric series?
5. How do you simplify nested square roots?

### Tip:
Always check the domain of a function to ensure no division by zero or undefined behavior for certain values (like \(x = 1\) in this case).

Calculate C, where C = 60 * sqrt{(1 + x + x^2 + ... + x^60) / (1 + x^-1 + x^-2 + ... + x^-60)} for x in positive real numbers.

Explore the computation of C using geometric series in the numerator and denominator, expressed with powers from 0 to 60 and inverse powers from -1 to -60, respectively. This detailed guide offers insights into series summation, limits, and their application in complex algebraic expressions. Ideal for university-level mathematics.

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