Sum to infinity of the series `2/3 - 5/6 + 2/3 -11/24` +….. Is

Options

A. `4/9`
B. `1/3`
C. `2/9`
D. `2/3`

Answer

To find the sum to infinity of the series \( S = \frac{2}{3} - \frac{5}{6} + \frac{2}{3} - \frac{11}{24} + \ldots \), we will follow these steps:

### Step 1: Define the series
Let \( S = \frac{2}{3} - \frac{5}{6} + \frac{2}{3} - \frac{11}{24} + \ldots \)

### Step 2: Multiply the series by \(-\frac{1}{2}\)
We will multiply both sides of the equation by \(-\frac{1}{2}\):
\[ -\frac{S}{2} = -\frac{2}{6} + \frac{5}{12} - \frac{8}{24} + \ldots \]

### Step 3: Rewrite the equations
Now we have two equations:
1. \( S = \frac{2}{3} - \frac{5}{6} + \frac{2}{3} - \frac{11}{24} + \ldots \) (Equation 1)
2. \( -\frac{S}{2} = -\frac{2}{6} + \frac{5}{12} - \frac{8}{24} + \ldots \) (Equation 2)

### Step 4: Subtract Equation 2 from Equation 1
Now, we will subtract Equation 2 from Equation 1:
\[ S + \frac{S}{2} = \left(\frac{2}{3} - \frac{5}{6} + \frac{2}{3} - \frac{11}{24} + \ldots\right) - \left(-\frac{2}{6} + \frac{5}{12} - \frac{8}{24} + \ldots\right) \]
This simplifies to:
\[ \frac{3S}{2} = \frac{2}{3} - \left(-\frac{5}{6} + \frac{2}{6} + \frac{5}{12} - \frac{8}{24} + \ldots\right) \]

### Step 5: Combine like terms
Now, we will combine the terms on the right-hand side:
\[ \frac{3S}{2} = \frac{2}{3} - \left(-\frac{3}{6} + \frac{5}{12} + \frac{3}{24} + \ldots\right) \]

### Step 6: Identify the geometric series
The series on the right can be recognized as a geometric series:
\[ \frac{3}{6} = \frac{1}{2}, \quad \frac{5}{12} = \frac{5}{12}, \quad \frac{3}{24} = \frac{1}{8}, \ldots \]
This series has a first term \( a = \frac{1}{2} \) and a common ratio \( r = -\frac{1}{2} \).

### Step 7: Use the formula for the sum of an infinite geometric series
The sum of an infinite geometric series is given by:
\[ \text{Sum} = \frac{a}{1 - r} \]
Thus, substituting \( a = \frac{1}{2} \) and \( r = -\frac{1}{2} \):
\[ \text{Sum} = \frac{\frac{1}{2}}{1 - (-\frac{1}{2})} = \frac{\frac{1}{2}}{1 + \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{3}{2}} = \frac{1}{3} \]

### Step 8: Substitute back into the equation
Now substituting back into our equation:
\[ \frac{3S}{2} = \frac{2}{3} - \frac{1}{3} \]
This simplifies to:
\[ \frac{3S}{2} = \frac{1}{3} \]

### Step 9: Solve for \( S \)
Now, multiply both sides by \( \frac{2}{3} \):
\[ S = \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{9} \]

### Final Answer
Thus, the sum to infinity of the series is:
\[ \boxed{\frac{2}{9}} \]