\( Q: \frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}+\cdots+\frac{1}{x^{12}} \) Find sam of the series

To find the sum of the series: \[ S = \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + \cdots + \frac{1}{x^{12}}, \] we can recognize this as a **geometric series**. A geometric series has the form: \[ S = a + ar + ar^{2} + \cdots + ar^{n-1}, \] where: - \( a \) is the first term, - \( r \) is the common ratio, - \( n \) is the number of terms. ### Step-by-Step Solution: 1. **Identify the Parameters:** - **First term (\( a \))**: \( \frac{1}{x} \) - **Common ratio (\( r \))**: \( \frac{1}{x} \) - **Number of terms (\( n \))**: 12 2. **Formula for the Sum of a Geometric Series:** The sum \( S \) of the first \( n \) terms of a geometric series is given by: \[ S = a \frac{1 - r^{n}}{1 - r} \] 3. **Plugging in the Values:** \[ S = \frac{1}{x} \cdot \frac{1 - \left(\frac{1}{x}\right)^{12}}{1 - \frac{1}{x}} \] 4. **Simplify the Expression:** Simplify the numerator and denominator: \[ S = \frac{1 - \frac{1}{x^{12}}}{x - 1} \] This simplification is achieved by multiplying the numerator and the denominator by \( x \), which eliminates the fractions. ### Final Answer: \[ \boxed{\,\frac{1 - \dfrac{1}{x^{12}}}{\,x\;-\;1\,}\,} \]