Sum up to n terms the series
`1+3x+5x^(2)+7x^(3)+...`

Answer

To find the sum up to \( n \) terms of the series \( S_n = 1 + 3x + 5x^2 + 7x^3 + \ldots \), we can derive a formula step by step.

### Step-by-Step Solution:

1. **Identify the Series**:
We have the series:
\[ S_n = 1 + 3x + 5x^2 + 7x^3 + \ldots + (2n-1)x^{n-1} \]
where the \( k \)-th term is given by \( (2k - 1)x^{k-1} \).

2. **Multiply the Series by \( x \)**:
We multiply the entire series by \( x \):
\[ x S_n = x + 3x^2 + 5x^3 + 7x^4 + \ldots + (2n-1)x^n \]

3. **Subtract the Two Equations**:
Now we subtract \( x S_n \) from \( S_n \):
\[ S_n - x S_n = (1 + 3x + 5x^2 + 7x^3 + \ldots + (2n-1)x^{n-1}) - (x + 3x^2 + 5x^3 + 7x^4 + \ldots + (2n-1)x^n) \]
This gives:
\[ S_n(1 - x) = 1 + (2x) + (2x^2) + (2x^3) + \ldots + (2x^{n-1}) - (2n-1)x^n \]

4. **Simplify the Series**:
The series \( 2x + 2x^2 + 2x^3 + \ldots + 2x^{n-1} \) can be factored:
\[ 2x(1 + x + x^2 + \ldots + x^{n-2}) = 2x \cdot \frac{1 - x^{n-1}}{1 - x} \]
Thus, we can rewrite the equation:
\[ S_n(1 - x) = 1 + 2x \cdot \frac{1 - x^{n-1}}{1 - x} - (2n-1)x^n \]

5. **Rearranging the Equation**:
Now we have:
\[ S_n(1 - x) = 1 + \frac{2x(1 - x^{n-1})}{1 - x} - (2n-1)x^n \]
Dividing both sides by \( 1 - x \):
\[ S_n = \frac{1}{1 - x} + \frac{2x(1 - x^{n-1})}{(1 - x)^2} - \frac{(2n-1)x^n}{1 - x} \]

6. **Final Expression for \( S_n \)**:
Thus, the sum of the series up to \( n \) terms is:
\[ S_n = \frac{1}{1 - x} + \frac{2x(1 - x^{n-1})}{(1 - x)^2} - \frac{(2n-1)x^n}{1 - x} \]