If `y= x^(1/3).x^(1/9).x^(1/27)...oo` then y=?

Answer

To solve the problem \( y = x^{1/3} \cdot x^{1/9} \cdot x^{1/27} \cdots \) up to infinity, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression for \( y \):
\[ y = x^{1/3} \cdot x^{1/9} \cdot x^{1/27} \cdots \]
We can express this as:
\[ y = x^{\left( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \right)} \]

### Step 2: Identify the series
The series in the exponent is:
\[ S = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \]
This is a geometric series where the first term \( a = \frac{1}{3} \) and the common ratio \( r = \frac{1}{3} \).

### Step 3: Sum the geometric series
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values of \( a \) and \( r \):
\[ S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \]

### Step 4: Substitute back into the expression for \( y \)
Now we can substitute \( S \) back into the expression for \( y \):
\[ y = x^{\frac{1}{2}} \]

### Final Answer
Thus, the final result is:
\[ y = \sqrt{x} \]

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