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Fill in the blanks to complete each of the following theorem statements.

Tests for Monotonicity: A sequence {ak} is increasing if it passes any of the following tests:

The Derivative Test: ____0forallx1 given that a(x) is a function that is _______on [1,) and whose value at any positive integer k isa(k)=____

Short Answer

Expert verified

The required answer is a'(x)0for all x1given that a(x) is a function that is differentiable on [1,)and whose value at any positive integer k isa(k)=ak

Step by step solution

01

Step 1. Given Information 

The given term is Derivative test.

02

Step 2. Explanation 

The derivative test is based on the test for increasing behavior in differentiable functions.

Thus,

a'(x)0forallx1given that a(x) is a function that is differentiable on [1,)and whose value at any positive integer k isa(k)=ak

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