Provide an Appropriate Response $\ln ( 1 + x )$ For $x > 1$

question 51

Multiple Choice

Provide an appropriate response.
-Derive a series for $\ln ( 1 + x )$ for $x > 1$ by first finding the series for $\frac { 1 } { 1 + x }$ and then integrating. $\left( \right.$ Hint: $\left. \frac { 1 } { 1 + x } = \frac { 1 } { x } \frac { 1 } { 1 + 1 / x } \right)$

• A) \(\ln x + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n x ^ { n - 1 } }\)
• B) \(\ln x + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n x ^ { n } }\)
• C) \(\ln x + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n x ^ { n - 1 } }\)
• D) \(\ln x + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n x ^ { n } }\)

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Definitions:

Ausgleich

The compromise of 1867 that established the dual monarchy of Austria-Hungary.

Dual Monarchy

The arrangement in which two separate kingdoms are ruled by the same monarch while maintaining distinct governments and institutions, as with Austria-Hungary from 1867 to 1918.

Austria-Hungary

A central European empire existing from 1867 to 1918, it was a dual monarchy consisting of Austria and Hungary, which dissolved after its defeat in World War I.