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

question 133

Multiple Choice

Use the binomial series to find the Maclaurin series for the function $f ( x ) = \frac { 1 } { \sqrt { 81 + x ^ { 2 } } }$

• A) \(\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { 9 ^ { n } n ! }\)
• B)\(\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } ( 2 n - 1 ) ! 9 ^ { 2 n + 1 } }\)
• C)\(\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } n ! 9 ^ { 2 n + 1 } }\)
• D)\(\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { n } } { n ! }\)
• E)\(\frac { 1 } { \sqrt { 81 + x ^ { 2 } } } = \frac { 1 } { 9 } + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 1 \cdot 3 \cdot 5 \cdots ( 2 n - 1 ) x ^ { 2 n } } { 2 ^ { n } 9 ^ { 2 n + 1 } }\)

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