If `x=1-y+y^(2)-y^(3)+.....` up to infinite terms, where `|y|lt1`, then which one of the following is correct?

Options

A. `x=(1)/(1+y)`
B. `x=(y)/(1-y)`
C. `x=(y)/(1+y)`
D. `x=(y)/(1-y)`

Explanation

To solve the problem, we need to find the sum of the infinite series given by:

\[ x = 1 - y + y^2 - y^3 + \ldots \]

This series can be recognized as a geometric series. The general form of a geometric series is:

\[ S = a + ar + ar^2 + ar^3 + \ldots \]

where:
- \( a \) is the first term,
- \( r \) is the common ratio.

In our series:
- The first term \( a = 1 \),
- The common ratio \( r = -y \).

Since the series converges for \( |y| < 1 \), we can use the formula for the sum of an infinite geometric series:

\[ S = \frac{a}{1 - r} \]

Substituting our values into the formula:

1. Identify \( a \) and \( r \):
- \( a = 1 \)
- \( r = -y \)

2. Substitute into the sum formula:
\[ x = \frac{1}{1 - (-y)} = \frac{1}{1 + y} \]

Thus, we find that:

\[ x = \frac{1}{1 + y} \]

Now, we can summarize the steps:

1. Recognize the series as a geometric series.
2. Identify the first term \( a \) and the common ratio \( r \).
3. Use the formula for the sum of an infinite geometric series.
4. Substitute the values of \( a \) and \( r \) into the formula to find \( x \).