Provide an Appropriate Response $\frac { x } { 1 + x ^ { 2 } }$

question 95

Multiple Choice

Provide an appropriate response.
-Derive the series for $\frac { x } { 1 + x ^ { 2 } }$ for $x > 1$ by first writing $\frac { x } { 1 + x ^ { 2 } } = \frac { 1 } { x } \frac { 1 } { 1 + 1 / x ^ { 2 } }$ .

• A) \(\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { x ^ { 2 n } }\)
• B) \(\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { x ^ { 2 n + 1 } }\)
• C) \(\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { x ^ { 2 n + 1 } }\)
• D) \(\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { x ^ { 2 n } }\)

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Definitions:

pH

A scale measuring the acidity or alkalinity of a solution, where 7 is neutral, values below 7 are acidic, and values above 7 are alkaline.

Buffers

Substances that minimize changes in the pH of a solution by absorbing or releasing hydrogen ions.

pH

A logarithmic scale used to specify the acidity or basicity of an aqueous solution, ranging from 0 (very acidic) to 14 (very basic) with 7 being neutral.

Hydrogen Ions

Positively charged ions (H+) that contribute to acidity or alkalinity of solutions and play critical roles in many biological processes.