Find the terms of the Maclaurin series for $f ( x ) = \frac { 1 } { \sqrt { 1 + 2 x } }$ , as far as the term in $x ^ { 3 }$ .
A) $1 - x + x ^ { 2 } - x ^ { 3 }$
B) $1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 }$
C) $1 - x + \frac { 3 } { 2 } x ^ { 2 } - \frac { 5 } { 2 } x ^ { 3 }$
D) $1 + x + 3 x ^ { 2 } + 5 x ^ { 3 }$
E) $1 - x + \frac { 3 } { 2 } x ^ { 2 } - \frac { 7 } { 3 } x ^ { 3 }$
F) $1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 7 } { 3 } x ^ { 3 }$
G) $1 - x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 7 } { 3 } x ^ { 3 }$
H) $1 + x + \frac { 7 } { 2 } x ^ { 2 } + \frac { 11 } { 3 } x ^ { 3 }$

Question

Find the terms of the Maclaurin series for $f ( x ) = \frac { 1 } { \sqrt { 1 + 2 x } }$ , as far as the term in $x ^ { 3 }$ .
A) $1 - x + x ^ { 2 } - x ^ { 3 }$ 
B) $1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 }$ 
C) $1 - x + \frac { 3 } { 2 } x ^ { 2 } - \frac { 5 } { 2 } x ^ { 3 }$ 
D) $1 + x + 3 x ^ { 2 } + 5 x ^ { 3 }$ 
E) $1 - x + \frac { 3 } { 2 } x ^ { 2 } - \frac { 7 } { 3 } x ^ { 3 }$ 
F) $1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 7 } { 3 } x ^ { 3 }$ 
G) $1 - x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 7 } { 3 } x ^ { 3 }$ 
H) $1 + x + \frac { 7 } { 2 } x ^ { 2 } + \frac { 11 } { 3 } x ^ { 3 }$

Quizplus · Accepted Answer

The Maclaurin series for $f(x) = \frac{1}{\sqrt{1 + 2x}}$ can be derived from the binomial series expansion for $(1 + x)^n$, where $n = -\frac{1}{2}$ and $x$ is replaced by $2x$. The series is $1 - x + \frac{3}{2}x^2 - \frac{5}{2}x^3 + \cdots$, which matches option C.

Find the Terms of the Maclaurin Series For f(x)=11+2xf ( x ) = \frac { 1 } { \sqrt { 1 + 2 x } }f(x)=1+2x​1​

Find the Terms of the Maclaurin Series For $f ( x ) = \frac { 1 } { \sqrt { 1 + 2 x } }$