Home » Blog » Prove or disprove: If lim f(x) = 0 and lim In = A then ∫ f(x) converges to A

Prove or disprove: If lim f(x) = 0 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.

Assume

    \[ \lim_{x \to \infty} f(x) = 0 \qquad \text{and} \qquad \lim_{n \to \infty} I_n = A. \]

Then

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


Incomplete.

2 comments

  1. Evangelos says:

    Proof. For some positive real x such that n \leq x \leq n+1, we have

        \begin{align*} \int_{1}^{x} f(x)\, dx &= \int_{1}^{n} f(x)\, dx + \int_{n}^{x} f(x)\, dx \\ &= I_{n} + (F(x) - I_{n}) \end{align*}

    where F(x) = \int_{1}^{x} f(x)\, dx. But as n \rightarrow \infty and hence x \rightarrow \infty, f(x) \rightarrow 0, and thus F(x) - I_{n} = \int_{n}^{x} f(x)\, dx \rightarrow 0. But we know \lim_{n \to \infty} I_{n} = A, which implies \lim_{x \to \infty} F(x) = A. Thus, we have shown that

        \begin{align*} \lim_{x \to \infty} \int_{1}^{x} f(x)\, dx = A  \end{align*}

    This completes the proof.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):