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

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UpStudy AI · Accepted Answer

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:

$$
\boxed{\,\frac{1 - \dfrac{1}{x^{12}}}{\,x\;-\;1\,}\,}
$$
The sum of the series is $$ \frac{1 - \frac{1}{x^{12}}}{x - 1} $$.

Find sam of the series

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