If ` e^(e^(x)) = a_(0) +a_(1)x +a_(2)x^(2)+...` ,then find the value of a_ (0)

Options

A. `a_(0)=1`
B. `a_(0)=e`
C. `a_(0)=e^(e )`
D. `a_(0)=e^(2)`

Explanation

To find the value of \( a_0 \) in the series expansion of \( e^{e^x} \), we will follow these steps:

### Step 1: Understand the function
The function given is \( e^{e^x} \). We need to express this in a power series form.

### Step 2: Expand \( e^x \)
The Taylor series expansion of \( e^x \) around \( x = 0 \) is:
\[ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \]

### Step 3: Substitute \( e^x \) into \( e^{e^x} \)
Now, we substitute \( e^x \) into the exponential function:
\[ e^{e^x} = e^{1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots} \]

### Step 4: Use the series expansion for \( e^u \)
The series expansion for \( e^u \) is:
\[ e^u = 1 + \frac{u}{1!} + \frac{u^2}{2!} + \frac{u^3}{3!} + \ldots \]
where \( u = e^x \).

### Step 5: Find \( a_0 \)
To find \( a_0 \), we evaluate \( e^{e^0} \):
\[ e^{e^0} = e^1 = e \]
Thus, the constant term \( a_0 \) in the series expansion is:
\[ a_0 = e \]

### Final Answer
The value of \( a_0 \) is \( e \).

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