Provide an Appropriate Response For $| x | < 1$ To Find the Series For

question 178

Multiple Choice

Provide an appropriate response.
-Use the fact that $\sin ^ { - 1 } x = x + \sum _ { n = 1 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdot \ldots \cdot ( 2 n - 1 ) x ^ { 2 n + 1 } } { 2 \cdot 4 \cdot 6 \cdot \ldots \cdot ( 2 n ) ( 2 n + 1 ) }$ for $| x | < 1$ to find the series for $\cos ^ { - 1 } x$ .

• A) \(\frac { \pi } { 2 } - x + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \cdot 1 \cdot 3 \cdot 5 \cdot \ldots \cdot ( 2 n - 1 ) x ^ { 2 n + 1 } } { 2 \cdot 4 \cdot 6 \cdot \ldots \cdot ( 2 n ) ( 2 n + 1 ) }\)
• B) \(\frac { \pi } { 2 } - x - \sum _ { n = 1 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdot \ldots \cdot ( 2 n - 1 ) x ^ { 2 n + 1 } } { 2 \cdot 4 \cdot 6 \cdot \ldots \cdot ( 2 n ) ( 2 n + 1 ) }\)
• C) \(\frac { \pi } { 2 } - \sum _ { n = 1 } ^ { \infty } \frac { 1 \cdot 3 \cdot 5 \cdot \ldots \cdot ( 2 n - 1 ) x ^ { 2 n + 1 } } { 2 \cdot 4 \cdot 6 \cdot \ldots \cdot ( 2 n ) ( 2 n + 1 ) }\)
• D) \(\frac { \pi } { 2 } - x + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } \cdot 1 \cdot 3 \cdot 5 \cdot \ldots \cdot ( 2 n - 1 ) x ^ { 2 n + 1 } } { 2 \cdot 4 \cdot 6 \cdot \ldots \cdot ( 2 n ) ( 2 n + 1 ) }\)

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