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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.

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