The coefficient of `x^8` in the expasnsion of `(1+(x^2)/(2!)+(x^4)/(4!)+(x^6)/(6!)+(x^8)/(8!))^2` is

Options

A. `1/135`
B. `2/315`
C. `3/105`
D. `1/210`

Answer

To find the coefficient of \( x^8 \) in the expansion of

\[ \left(1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \frac{x^8}{8!}\right)^2, \]

we can follow these steps:

### Step 1: Recognize the series
The expression inside the parentheses resembles the Taylor series expansion of \( e^x \), but only includes even powers of \( x \). This can be interpreted as:

\[ \frac{1}{2} \left( e^x + e^{-x} \right) = \cosh(x). \]

Thus, we can rewrite our expression as:

\[ \left( \cosh(x) \right)^2. \]

### Step 2: Use the identity for hyperbolic cosine
Using the identity \( \cosh^2(x) = \frac{1 + \cosh(2x)}{2} \), we can express our function as:

\[ \cosh^2(x) = \frac{1 + \cosh(2x)}{2}. \]

### Step 3: Expand \( \cosh(2x) \)
The expansion of \( \cosh(2x) \) is given by:

\[ \cosh(2x) = \sum_{n=0}^{\infty} \frac{(2x)^{2n}}{(2n)!} = \sum_{n=0}^{\infty} \frac{2^{2n} x^{2n}}{(2n)!}. \]

### Step 4: Find the coefficient of \( x^8 \)
Now, we need the coefficient of \( x^8 \) in the expansion of \( \frac{1 + \cosh(2x)}{2} \).

1. The constant term \( 1 \) contributes \( 0 \) to the coefficient of \( x^8 \).
2. The term \( \cosh(2x) \) contributes:

\[ \frac{(2x)^8}{8!} = \frac{2^8 x^8}{8!}. \]

Thus, the coefficient of \( x^8 \) in \( \cosh(2x) \) is \( \frac{2^8}{8!} \).

### Step 5: Combine results
The coefficient of \( x^8 \) in \( \cosh^2(x) \) is then:

\[ \frac{1}{2} \cdot \frac{2^8}{8!} = \frac{2^7}{8!}. \]

### Step 6: Simplify the expression
Calculating \( 2^7 = 128 \) and \( 8! = 40320 \):

\[ \frac{128}{40320}. \]

### Step 7: Simplify further
To simplify \( \frac{128}{40320} \):

1. Divide both numerator and denominator by \( 16 \):

\[ \frac{8}{2520}. \]

2. Divide again by \( 4 \):

\[ \frac{2}{630}. \]

3. Finally, divide by \( 2 \):

\[ \frac{1}{315}. \]

Thus, the coefficient of \( x^8 \) in the expansion is:

\[ \frac{1}{315}. \]

### Final Answer
The coefficient of \( x^8 \) in the expansion is \( \frac{1}{315} \).

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