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Prove or disprove: If f is positive and lim In = A, then ∫ f(x) converges to A

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 positive and if

    \[ \lim_{n \to \infty} I_n = A \]


    \[ \int_1^{\infty} f(x) \, dx = A. \]



  1. Evangelos says:

    After looking at the counterexample for exercise 25, I’ll have to rework this one because f \rightarrow 0 is not a necessary condition for convergence. There’s actually a pretty simple proof for this one since f is set to be positive for all x

    Proof. If f(x) is positive for all x, then, for any x \leq n for some integer n \geq 1, the integral \int_{1}{x} f(t) \, dt is bounded above by I_{n}. Thus, if I_{n} \rightarrow A as n \rightarrow \infty, then so does \int_{1}{x} f(t) \, dt.

  2. Evangelos says:

    Proof. Since f(x) takes positive values for all x \geq 1, for I_{n} to converge as n \rightarrow \infty, f(x) must go to zero as x \rightarrow \infty. But if f(x) \rightarrow 0 as x \rightarrow \infty and \lim_{n \to \infty} I_{n} = A, then, following the proof written in exercise 21,

        <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-632d7d4576d4330cae95d752743cbe38_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"/>

    converges and has value A.

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