Comparison Theorem

Tags

Calculus

Cegep/2

Word count

79 words

Reading time

1 minute

Theorem that determines convergence of a function by comparing it to another function of known convergence

Suppose that $f$ and $g$ are functions such that $f(x)\ge g(x)\ge 0$ for $x\in [a,\mathrm{\infty })$.
Then,

1. If ${\int }_a^{\mathrm{\infty }}f(x)\mathrm{d}x$ is convergent, then ${\int }_a^{\mathrm{\infty }}g(x)\mathrm{d}x$ is also convergent.
2. If ${\int }_a^{\mathrm{\infty }}g(x)\mathrm{d}x$ is divergent, then ${\int }_a^{\mathrm{\infty }}f(x)\mathrm{d}x$ is also divergent.

Contributors

Ajitani Hifumi

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