The largest interval for which the series `1+(x-1)+(x-1)^(2)+….oo` may be summed is

Options

A. `0 lt x lt 1`
B. `0 lt x lt 2`
C. `-1 lt x lt 1`
D. ` -2 lt x lt 2 `

Answer

To find the largest interval for which the series \( 1 + (x-1) + (x-1)^2 + \ldots \) can be summed, we will analyze the series step by step.

### Step 1: Identify the Series
The series can be expressed as:
\[ S = 1 + (x-1) + (x-1)^2 + \ldots \]
This is a geometric series where the first term \( a = 1 \) and the common ratio \( r = x - 1 \).

### Step 2: Condition for Convergence
For a geometric series to converge, the absolute value of the common ratio must be less than 1:
\[ |r| < 1 \]
Thus, we have:
\[ |x - 1| < 1 \]

### Step 3: Solve the Inequality
The inequality \( |x - 1| < 1 \) can be broken down into two cases:
1. \( x - 1 < 1 \)
2. \( -(x - 1) < 1 \) (which simplifies to \( x - 1 > -1 \))

#### Case 1: \( x - 1 < 1 \)
\[ x < 2 \]

#### Case 2: \( x - 1 > -1 \)
\[ x > 0 \]

### Step 4: Combine the Results
From both cases, we have:
\[ 0 < x < 2 \]
This means the largest interval for which the series can be summed is:
\[ (0, 2) \]

### Conclusion
The largest interval for which the series \( 1 + (x-1) + (x-1)^2 + \ldots \) can be summed is:
\[ \boxed{(0, 2)} \]