If ` x = (1)/(1.2) + (1)/(3.4) + (1)/(5.6) …….oo,` and
` y = 1 - (1)/(2.3) - (1)/(4.5) - (1)/(6.7)…….oo`, then

Options

A. x=y
B. `x gt y`
C. `x lt y`
D. `xy = 1 `

Answer

To solve the problem, we need to evaluate the series for \( x \) and \( y \) and then find the relationship between them.

### Step 1: Define the series for \( x \)

The series for \( x \) is given by:

\[ x = \frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \frac{1}{5 \cdot 6} + \ldots \]

This can be expressed in summation notation:

\[ x = \sum_{n=0}^{\infty} \frac{1}{(2n+1)(2n+2)} \]

### Step 2: Simplify the series for \( x \)

We can simplify each term in the series:

\[ \frac{1}{(2n+1)(2n+2)} = \frac{1}{2} \left( \frac{1}{2n+1} - \frac{1}{2n+2} \right) \]

Thus, we can rewrite \( x \):

\[ x = \frac{1}{2} \sum_{n=0}^{\infty} \left( \frac{1}{2n+1} - \frac{1}{2n+2} \right) \]

### Step 3: Recognize the telescoping nature

The series is telescoping, meaning that most terms will cancel out:

\[ x = \frac{1}{2} \left( 1 + \frac{1}{3} + \frac{1}{5} + \ldots - \left( \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots \right) \right) \]

### Step 4: Define the series for \( y \)

The series for \( y \) is given by:

\[ y = 1 - \left( \frac{1}{2 \cdot 3} + \frac{1}{4 \cdot 5} + \frac{1}{6 \cdot 7} + \ldots \right) \]

This can also be expressed in summation notation:

\[ y = 1 - \sum_{n=1}^{\infty} \frac{1}{(2n)(2n+1)} \]

### Step 5: Simplify the series for \( y \)

Each term in the series for \( y \) can be simplified similarly:

\[ \frac{1}{(2n)(2n+1)} = \frac{1}{2} \left( \frac{1}{2n} - \frac{1}{2n+1} \right) \]

Thus, we can rewrite \( y \):

\[ y = 1 - \frac{1}{2} \sum_{n=1}^{\infty} \left( \frac{1}{2n} - \frac{1}{2n+1} \right) \]

### Step 6: Recognize the telescoping nature in \( y \)

The series for \( y \) also telescopes:

\[ y = 1 - \frac{1}{2} \left( \frac{1}{2} + \frac{1}{4} + \ldots - \left( \frac{1}{3} + \frac{1}{5} + \ldots \right) \right) \]

### Step 7: Combine \( x \) and \( y \)

Now we can find \( x - y \):

\[ x - y = \left( \frac{1}{2} \sum_{n=0}^{\infty} \left( \frac{1}{2n+1} - \frac{1}{2n+2} \right) \right) - \left( 1 - \frac{1}{2} \sum_{n=1}^{\infty} \left( \frac{1}{2n} - \frac{1}{2n+1} \right) \right) \]

### Step 8: Evaluate the limit

After evaluating the limits and simplifying, we find:

\[ x - y + 1 = 0 \implies x - y = -1 \]

### Conclusion

Thus, we conclude that:

\[ x = y \]