Cakculate the sum of the series \[ x^{5}-x^{3}+x \cdots+\frac{1}{x^{11}} \]

To calculate the sum of the series \( S = x^{5} - x^{3} + x - \frac{1}{x} + \frac{1}{x^{3}} - \frac{1}{x^{5}} + \frac{1}{x^{7}} - \frac{1}{x^{9}} + \frac{1}{x^{11}} \), we can group the terms based on their powers and simplify. First, rewrite the series: \[ S = x^{5} - x^{3} + x - \frac{1}{x} + \frac{1}{x^{3}} - \frac{1}{x^{5}} + \frac{1}{x^{7}} - \frac{1}{x^{9}} + \frac{1}{x^{11}} \] Now we realize that this series can be represented as a combination of two parts: one for positive powers of \( x \) and the other for negative powers of \( x \). Rearranging: \[ S = (x^{5} - x^{3} + x) + \left( -\frac{1}{x} + \frac{1}{x^{3}} - \frac{1}{x^{5}} + \frac{1}{x^{7}} - \frac{1}{x^{9}} + \frac{1}{x^{11}} \right) \] Next, recognize that the terms \( +\frac{1}{x^{7}} - \frac{1}{x^{9}} + \frac{1}{x^{11}} \) can follow the pattern of a geometric series with a slight adjustment using negative powers: \[ \frac{1}{x} + 0 \left( - \frac{1}{x^{3}} + \frac{1}{x^{5}} + \cdots \right) \] We can find the simplification by calculating the positive and negative parts separately. The sum of the positive power series \( x^{5} - x^{3} + x \) can be evaluated as: \[ x + (-x^{3} + x^{5}) = x + (\text{terms in descending order}) \] Now, the negative power part can be considered as: \[ -\left( \frac{1}{x} + \frac{1}{x^{3}} - \frac{1}{x^{5}} + \frac{1}{x^{7}} - \frac{1}{x^{9}} + \frac{1}{x^{11}} \right) \] Using the knowledge of geometric series: \[ \sum_{k=0}^{n} ar^k = \frac{a(1 - r^{n+1})}{1 - r} \] The overall sum \( S \) can be computed efficiently using the recognized pattern. Thus, the complete sum of the series would require substituting in values for \( x \) or \( \frac{1}{x} \) as per the series rules specified.