• Title: Comparison Test for Integrals

  • Series: Real Analysis

  • Chapter: Riemann Integral

  • YouTube-Title: Real Analysis 61 | Comparison Test for Integrals

  • Bright video: https://youtu.be/yEp9BTDgOjk

  • Dark video: https://youtu.be/WoiYPegVj6M

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ra61_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $f,g: [0,\infty) \rightarrow [0,\infty)$ be functions such that the restrictions to compact intervals are Riemann-integrable. Which claim is correct?

    A1: If $\int_0^{\infty} g(x) dx$ converges, then $\int_0^{\infty} f(x) dx$ converges.

    A2: If $\int_0^{\infty} g(x) dx$ converges and $f \leq g$, then $\int_0^{\infty} f(x) dx$ converges.

    A3: If $\int_0^{\infty} g(x) dx$ converges and $g \leq f$, then $\int_0^{\infty} f(x) dx$ converges.

    A4: If $\int_0^{\infty} g(x) dx$ diverges and $g \leq f$, then $\int_0^{\infty} f(x) dx$ converges.

    Q2: Let $f,g: [0,\infty) \rightarrow [0,\infty)$ be functions such that the restrictions to compact intervals are Riemann-integrable. Which claim is correct?

    A1: If $\int_0^{\infty} g(x) dx$ diverges, then $\int_0^{\infty} f(x) dx$ diverges.

    A2: If $\int_0^{\infty} g(x) dx$ diverges and $f \leq g$, then $\int_0^{\infty} f(x) dx$ diverges.

    A3: If $\int_0^{\infty} g(x) dx$ diverges and $g \leq f$, then $\int_0^{\infty} f(x) dx$ diverges.

    A4: If $\int_0^{\infty} g(x) dx$ diverges and $g \leq f$, then $\int_0^{\infty} f(x) dx$ converges.

    Q3: Is the integral $\int_1^{\infty} \frac{1-x}{x^2} dx$ convergent?

    A1: Yes!

    A2: No!

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