In the text, we showed you one way to verify that a sequence...

In the text, we showed you one way to verify that a sequence is decreasing. As an alternative, explain why if $a_{k}=f(k)$ and $f^{\prime}(k)<0,$ then the sequence $a_{k}$ is decreasing. Use this method to prove that $a_{k}=\frac{k}{k^{2}+2}$ is decreasing.

Key Concepts

Application of the Quotient Rule in Differentiation

In order to verify that a function, especially a rational function, has a negative derivative, the quotient rule is a useful technique. The quotient rule provides a systematic way to differentiate a function defined as one function divided by another. By applying the quotient rule, one can simplify the expression and determine the sign of the derivative, which in turn can confirm whether the related sequence is decreasing.

Connecting Continuous Functions and Sequences

Often, sequences are obtained as the discrete evaluation of continuous functions. By analyzing the derivative of the underlying continuous function, one can infer the behavior of the sequence. This approach leverages the properties of differentiable functions—namely, that a continuous function with a negative derivative is decreasing—to provide a straightforward method for proving the monotonicity of sequences.

Derivative Test for Monotonicity

The derivative test for monotonicity is a fundamental calculus concept stating that if a function’s derivative is negative throughout an interval, then the function is strictly decreasing on that interval. When a sequence is defined by a function evaluated at integer values, i.e., a? = f(k), if f?(x) is negative for all relevant x, it follows that f(k + 1) < f(k), hence the sequence is decreasing.

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