The co-efficient of `x^(401)` in the expansion of `(1+x+x^(2)+.....+x^(9))^(-1),(|x| lt 1)` is

Options

A. 1
B. `-1`
C. 2
D. `-2`

Answer

To find the coefficient of \( x^{401} \) in the expansion of \( (1 + x + x^2 + \ldots + x^9)^{-1} \), we can follow these steps:

### Step 1: Simplify the expression
The series \( 1 + x + x^2 + \ldots + x^9 \) is a geometric series. The sum of a geometric series can be expressed as:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a = 1 \), \( r = x \), and there are \( 10 \) terms (from \( x^0 \) to \( x^9 \)). Thus, we have:
\[ 1 + x + x^2 + \ldots + x^9 = \frac{1(1 - x^{10})}{1 - x} = \frac{1 - x^{10}}{1 - x} \]

### Step 2: Substitute into the original expression
Now, we need to find:
\[ \left( \frac{1 - x^{10}}{1 - x} \right)^{-1} = \frac{1 - x}{1 - x^{10}} \]

### Step 3: Expand the expression
We can rewrite this as:
\[ (1 - x)(1 - x^{10})^{-1} \]
Using the binomial series expansion for \( (1 - x^{10})^{-1} \):
\[ (1 - x^{10})^{-1} = \sum_{n=0}^{\infty} x^{10n} \]
Thus,
\[ (1 - x)(1 - x^{10})^{-1} = (1 - x) \sum_{n=0}^{\infty} x^{10n} = \sum_{n=0}^{\infty} x^{10n} - \sum_{n=0}^{\infty} x^{10n + 1} \]

### Step 4: Combine the series
The first series gives us terms of the form \( x^{10n} \) and the second series gives us terms of the form \( x^{10n + 1} \). Therefore, we can express this as:
\[ \sum_{n=0}^{\infty} x^{10n} - \sum_{n=0}^{\infty} x^{10n + 1} = x^0 + x^{10} + x^{20} + \ldots - (x^1 + x^{11} + x^{21} + \ldots) \]

### Step 5: Identify the coefficient of \( x^{401} \)
To find the coefficient of \( x^{401} \), we note that:
- The term \( x^{401} \) can come from \( x^{10n} \) when \( 10n = 401 \), which is not possible since \( n \) must be an integer.
- The term \( x^{401} \) can also come from \( x^{10n + 1} \) when \( 10n + 1 = 401 \), leading to \( 10n = 400 \) or \( n = 40 \).

Thus, the coefficient of \( x^{401} \) in the expansion is \( -1 \) (from the second series).

### Final Answer
The coefficient of \( x^{401} \) in the expansion of \( (1 + x + x^2 + \ldots + x^9)^{-1} \) is \( -1 \).

---