The `3^(rd), 4^(th), 5^(th)` terms in the expansion of `log_(e )2` are respectively a, b, c then their ascending order is

Options

A. a, b, c
B. b, c, a
C. c, a, b
D. c, b, a

Answer

To solve the problem, we need to find the third, fourth, and fifth terms in the expansion of \( \log_e 2 \) using the series expansion for logarithms. The relevant series expansion is:

\[ \log(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \cdots \]

### Step 1: Identify the series expansion
We will use the series expansion of \( \log(1 + x) \) for \( x = 1 \):

\[ \log(1 + 1) = \log(2) = 1 - \frac{1^2}{2} + \frac{1^3}{3} - \frac{1^4}{4} + \frac{1^5}{5} - \cdots \]

### Step 2: Write out the terms
Now we can write out the first few terms of the series:

1. First term: \( 1 \)
2. Second term: \( -\frac{1}{2} \)
3. Third term: \( \frac{1}{3} \) (this is \( a \))
4. Fourth term: \( -\frac{1}{4} \) (this is \( b \))
5. Fifth term: \( \frac{1}{5} \) (this is \( c \))

### Step 3: Assign values to a, b, and c
From the above terms, we can assign:

- \( a = \frac{1}{3} \)
- \( b = -\frac{1}{4} \)
- \( c = \frac{1}{5} \)

### Step 4: Arrange in ascending order
Now we need to arrange \( a, b, c \) in ascending order:

- \( b = -\frac{1}{4} \) (smallest)
- \( c = \frac{1}{5} \)
- \( a = \frac{1}{3} \) (largest)

### Final Result
Thus, the ascending order of \( a, b, c \) is:

\[ b < c < a \]