Find the Taylor series at x = 0 of the function f(x) = x ln(1 + 2x) by computing three or four derivatives and using the definition of the Taylor series.
A) 2 $x ^ { 2 }$ - $\frac { 4 x ^ { 3 } } { 2 }$
B) 2 $x ^ { 2 }$ - $\frac { 4 x ^ { 3 } } { 3 }$
C) $2 x ^ { 2 }$ + $\frac { 4 x ^ { 3 } } { 2 ! }$
D) 2x - $\frac { 4 x ^ { 2 } } { 2 ! }$ + $\frac { 8 x ^ { 3 } } { 3 ! }$
E) none of these

Question

Quizplus · Accepted Answer

The Taylor series at x = 0 of the function f(x) = x ln(1 + 2x) is given by$$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$where$$f(0) = 0$$$$f'(x) = \frac{1 + 2x}{1 + 2x} + 2x \ln(1 + 2x) = 1 + 2x \ln(1 + 2x)$$$$f'(0) = 1$$$$f''(x) = 2 \ln(1 + 2x) + 4x \frac{1}{1 + 2x} = 2 \ln(1 + 2x) + \frac{4x}{1 + 2x}$$$$f''(0) = 2 \ln(1) = 0$$$$f'''(x) = \frac{4}{1 + 2x} - 8x \frac{1}{(1 + 2x)^2} = \frac{4}{1 + 2x} - \frac{8x}{(1 + 2x)^2}$$$$f'''(0) = 4$$Therefore, the Taylor series at x = 0 of the function f(x) = x ln(1 + 2x) is$$f(x) = x - \frac{4x^3}{3} + \cdots$$So, the correct choice is A.

Find the Taylor Series at X = 0 of the Function