Solve the Problem $f ( x ) = \sqrt [ a ] { x } , a > 0$

question 242

Essay

Solve the problem.
-Apply Newton's method to $f ( x ) = \sqrt [ a ] { x } , a > 0$ , and write an expression for $x _ { n } + 1$ . If the initial guess $x _ { 0 }$ is greater than or equal to 1 , what happens to $\left| x _ { n } + 1 \right|$ as $n \rightarrow \infty$ ?

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Definitions:

Phenylketonuria

A genetic disorder characterized by the body's inability to metabolize the amino acid phenylalanine, leading to its accumulation and potential brain damage.

Schizophrenia

A severe mental disorder characterized by disruptions in thought processes, perceptions, emotional responsiveness, and social interactions.

Bipolar Disorders

Mental health conditions characterized by extreme mood swings, including emotional highs (mania or hypomania) and lows (depression).

Intelligence

The ability to learn from experience, adapt to new situations, understand complex ideas, and utilize knowledge to manipulate one’s environment.