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

The following function f is defined for all x \geq 1, and n is a positive integer. Prove or provide a counterexample to the following statement.

Assume f'(x) exists for all x \geq 1 and is bounded,

    \[ |f'(x)| \leq M \]

for some constant M for all x \geq 1. Then,

    \[ \lim_{n \to \infty} I_n = A \quad \implies \quad \int_1^{\infty} f(x) \, dx  = A. \]


Incomplete.

One comment

  1. Evangelos says:

    Counterexample. Let f(x) &= \cos{2\pi x}. Then, f'(x) &= -2\pi \sin{2\pi x}, and \left \lvert f'(x) \right \rvert \leq 2\pi for all x \geq 1. From a prior counterexample (exercise 22), we know that although the sequence \{I_{n}\} converges to zero, the infinite integral

        <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a6a8bcc6190722c303952a14c33830aa_l3.png" height="36" width="81" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\int_{1}^{\infty} f(x) \, dx \]" title="Rendered by QuickLaTeX.com"/>

    diverges.

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