Question

Show that $a_{n}=\sqrt[3]{n+1}-n$ is decreasing.

   Show that $a_{n}=\sqrt[3]{n+1}-n$ is decreasing.

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 10, Problem 74

Instant Answer

Solved by Expert Fuzail Shakir

07/11/2019

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This function represents the sequence $a_{n}$ where $x=n$. Show more…

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Show that $a_{n}=\sqrt[3]{n+1}-n$ is decreasing.

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Key Concepts

Monotonic Sequence

A monotonic sequence is a sequence that consistently increases or decreases. In this context, showing that the sequence is decreasing means proving that each term is less than the previous term. This concept is fundamental in analysis and helps in understanding the behavior and convergence properties of sequences.

Difference Method for Sequences

This method involves examining the difference between successive terms, a??? - a?, to determine whether the sequence is increasing or decreasing. If the difference is negative for all valid n, it establishes that the sequence is strictly decreasing. This approach is a typical technique in proving monotonicity for sequences.

Properties of the Cube Root Function

The cube root function is strictly increasing, meaning that if one number is less than another, its cube root is also less. This inherent property is crucial when analyzing expressions involving cube roots, as it ensures that the ordering of numbers is preserved, and it plays a key role in comparing terms in a sequence defined using cube roots.

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