Sum up to n terms the series
`1-(2)/(5)+(3)/(5^(2))-(4)/(5^(3))`+ ...

Answer

To find the sum of the series \( S = 1 - \frac{2}{5} + \frac{3}{5^2} - \frac{4}{5^3} + \ldots \) up to \( n \) terms, we can follow these steps:

### Step 1: Identify the series
The series can be expressed as:
\[ S_n = \sum_{k=1}^{n} (-1)^{k-1} \frac{k}{5^{k-1}} \]

### Step 2: Write the series in a manageable form
We can rewrite the series as:
\[ S_n = 1 - \frac{2}{5} + \frac{3}{5^2} - \frac{4}{5^3} + \ldots + (-1)^{n-1} \frac{n}{5^{n-1}} \]

### Step 3: Divide the series by 5
Now, we will divide the entire series by 5:
\[ \frac{S_n}{5} = \frac{1}{5} - \frac{2}{5^2} + \frac{3}{5^3} - \frac{4}{5^4} + \ldots + (-1)^{n-1} \frac{n}{5^n} \]

### Step 4: Align the series
Align the terms of \( S_n \) and \( \frac{S_n}{5} \):
\[ S_n - \frac{S_n}{5} = 1 + \left(-\frac{2}{5} + \frac{1}{5}\right) + \left(\frac{3}{5^2} - \frac{2}{5^2}\right) + \left(-\frac{4}{5^3} + \frac{3}{5^3}\right) + \ldots + \left((-1)^{n-1} \frac{n}{5^{n-1}} - (-1)^{n-1} \frac{(n-1)}{5^n}\right) \]

### Step 5: Simplify the left-hand side
The left-hand side simplifies to:
\[ \frac{4S_n}{5} \]

### Step 6: Simplify the right-hand side
The right-hand side becomes:
\[ 1 + \left(-\frac{1}{5}\right) + \left(\frac{1}{5^2}\right) + \left(-\frac{1}{5^3}\right) + \ldots + \left((-1)^{n-1} \frac{1}{5^{n-1}}\right) \]

### Step 7: Recognize the geometric series
The series on the right is a geometric series with first term \( 1 \) and common ratio \( -\frac{1}{5} \):
\[ \text{Sum} = \frac{1 - (-\frac{1}{5})^n}{1 - (-\frac{1}{5})} = \frac{1 - (-\frac{1}{5})^n}{1 + \frac{1}{5}} = \frac{1 - (-\frac{1}{5})^n}{\frac{6}{5}} = \frac{5}{6} \left(1 - (-\frac{1}{5})^n\right) \]

### Step 8: Combine results
Thus, we have:
\[ \frac{4S_n}{5} = \frac{5}{6} \left(1 - (-\frac{1}{5})^n\right) \]

### Step 9: Solve for \( S_n \)
Multiplying both sides by \( \frac{5}{4} \):
\[ S_n = \frac{5}{4} \cdot \frac{5}{6} \left(1 - (-\frac{1}{5})^n\right) = \frac{25}{24} \left(1 - (-\frac{1}{5})^n\right) \]

### Final Result
Thus, the sum of the series up to \( n \) terms is:
\[ S_n = \frac{25}{24} \left(1 - (-\frac{1}{5})^n\right) \]