`1-(1)/(5)+(1.4)/(5.10) - (1.5.7)/(5.10.15)+…..=`

Options

A. `1/3 root(3)(5)`
B. `1/2 root(3)(5)`
C. `1/2root(3)(4)`
D. `root(3)(5)`

Answer

To solve the series \( S = 1 - \frac{1}{5} + \frac{1 \cdot 4}{5 \cdot 10} - \frac{1 \cdot 5 \cdot 7}{5 \cdot 10 \cdot 15} + \ldots \), we can analyze the pattern of the series and express it in a more manageable form.

### Step 1: Identify the terms of the series
The series can be rewritten as:
\[ S = 1 - \frac{1}{5} + \frac{1 \cdot 4}{5 \cdot 10} - \frac{1 \cdot 5 \cdot 7}{5 \cdot 10 \cdot 15} + \ldots \]

### Step 2: Recognize the pattern
The general term of the series can be expressed as:
\[ (-1)^n \frac{1 \cdot 4 \cdot (3n - 2)}{5^n \cdot n!} \]
where \( n \) starts from 0.

### Step 3: Relate the series to a known function
This series resembles the expansion of \( (1 - x)^{-k} \) for some \( k \). We can use the binomial series expansion:
\[ (1 - x)^{-k} = \sum_{n=0}^{\infty} \frac{(k+n-1)!}{(k-1)!n!} x^n \]

### Step 4: Find the appropriate \( k \) and \( x \)
In our case, we can relate:
\[ x = \frac{1}{5} \quad \text{and} \quad k = \frac{1}{3} \]

### Step 5: Substitute into the binomial expansion
Using the binomial expansion:
\[ (1 - x)^{-k} = (1 - \frac{1}{5})^{-\frac{1}{3}} = \left(\frac{4}{5}\right)^{-\frac{1}{3}} = \left(\frac{5}{4}\right)^{\frac{1}{3}} \]

### Step 6: Simplify the expression
Now we can express the series sum as:
\[ S = \left(\frac{5}{4}\right)^{\frac{1}{3}} \]

### Step 7: Final simplification
To express \( S \) in a more standard form:
\[ S = \frac{\sqrt[3]{5}}{2} \]

### Conclusion
Thus, the sum of the series is:
\[ S = \frac{1}{2} \sqrt[3]{5} \]

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