what does a function f(x)>=0f(x)>=0 for x>=0x>=0 satisfying ∫∞0f(x)=∞\int_0^{\infty}f(x)=\infty mean

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2 Answers

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I think it is clear what f(x)≥0f(x)\geq 0 for x≥0x\geq 0 means. The function’s values are greater than or equal 0 for all non-negative values of xx. So I assume the question you are asking is what exactly does

∫∞0f(x)dx=∞\int_0^\infty f(x)dx=\infty mean?

The proper way to understand this is as a limit:

∫∞0f(x)=lima→∞∫a0f(x)dx\int_0^\infty f(x)=\lim_{a\to\infty}\int_0^a f(x)dx This limit may or may not exist. The statement above that you asking about says that this limit does not exist: as aa grows, the definite integral from 00 to aa grows beyond any bound.

As an example, look at the function f(x)=\frac{1}{x+1}f(x)=\frac{1}{x+1}. You have

\int_0^a \frac{1}{x+1}dx = \log(a)-\log(1)=\log(a)\int_0^a \frac{1}{x+1}dx = \log(a)-\log(1)=\log(a)

The logarithm is a monotonously growing function that grows beyond any bound. So the limit of this integral as a\to\inftya\to\infty does not exist. This is being expressed by the integral being “infinity”.

In more advanced theories of integration based on measure theory there is a way for an integral to actually take the value of (positive or negative) infinity. The values of (real) integrals in such settings are taking their values in the “extended real numbers”. I am not sure this is what you are looking for though.

FANTASTIC first answer compared to nearly all I’ve seen. Welcome to the site Snake, and +1!

– Brevan Ellefsen

2 days ago

It means

\lim_{b\to\infty} \int_0^b f(x)\,dx = \infty

\lim_{b\to\infty} \int_0^b f(x)\,dx = \infty