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Prove or disprove: If f is monotonic decreasing and lim In exists then ∫ f(x) converges

by RoRi

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.

If f is monotonically decreasing and if

    \[ \lim_{n \to \infty} I_n \]

exists, then the improper integral

    \[ \int_1^{\infty} f(x) \, dx \]

converges.

Incomplete.

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