Use the Binomial Series to Find the Maclaurin Series for the Function

question 111

Multiple Choice

Use the binomial series to find the Maclaurin series for the function. $f ( x ) = \sqrt { 1 + x ^ { 2 } }$

• A) \(1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } - \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } + \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } - \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } + \cdots \cdots\)
• \[= 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 8 } x ^ { 4 } - \frac { 1 } { 16 } x ^ { 6 } + \frac { 5 } { 128 } x ^ { 8 } - \frac { 7 } { 256 } x ^ { 10 } + \cdots \cdots\]
• B)
• \[\begin{array} { l }
• 1 - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } - \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } - \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \\
• = 1 - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } - \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } - \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
• \end{array}\]
• C)
• \[\begin{array} { l }
• 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \\
• = 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
• \end{array}\]
• D)
• \[\begin{array} { l }
• 1 + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } + \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } + \cdots \cdots \\
• = 1 + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } + \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } + \cdots \cdots
• \end{array}\]
• E)
• \[\begin{array} { l }
• - 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 2 ! 2 ^ { 2 } } x ^ { 4 } + \frac { 3 } { 3 ! 2 ^ { 3 } } x ^ { 6 } - \frac { 3 ( 5 ) } { 4 ! 2 ^ { 4 } } x ^ { 8 } + \frac { 3 ( 5 ) ( 7 ) } { 5 ! 2 ^ { 5 } } x ^ { 10 } - \cdots \cdots \\
• = - 1 + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 8 } x ^ { 4 } + \frac { 1 } { 16 } x ^ { 6 } - \frac { 5 } { 128 } x ^ { 8 } + \frac { 7 } { 256 } x ^ { 10 } - \cdots \cdots
• \end{array}\]

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