The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

Assume exists for all and is bounded,

for some constant for all . Then,

**Incomplete.**

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Stumbling Robot

A Fraction of a Dot
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Prove or disprove a statement relating the derivative of a function to an improper integral of the function

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The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

Assume exists for all and is bounded,

for some constant for all . Then,

**Incomplete.**

Counterexample. Let . Then, , and for all . From a prior counterexample (exercise 22), we know that although the sequence converges to zero, the infinite integral

diverges.