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 solve the problem of finding the largest interval for which the series \(1 + (x-1) + (x-1)^2 + \ldots\) can be summed, we can follow these steps:

### Step 1: Identify the Series
The given series can be recognized as a geometric series:
\[ S = 1 + (x-1) + (x-1)^2 + \ldots \]

### Step 2: Determine the First Term and Common Ratio
In a geometric series, the first term \(a\) is:
\[ a = 1 \]
The common ratio \(r\) is:
\[ r = x - 1 \]

### Step 3: Condition for Convergence
For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1:
\[ |r| < 1 \]
This gives us:
\[ |x - 1| < 1 \]

### Step 4: Solve the Inequality
We can break down the absolute value inequality:
\[ -1 < x - 1 < 1 \]
Adding 1 to all parts of the inequality:
\[ 0 < x < 2 \]

### Step 5: Conclusion
The largest interval for which the series can be summed is:
\[ (0, 2) \]

### Step 6: Final Answer
Thus, the answer is:
\[ \text{The largest interval is } (0, 2). \]

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