Integral test - ultimately decreasing function

[TysonM8]

My textbook states that "For the integral test to apply, it is not necessary that f be always decreasing. What is important is that f be ultimately decreasing."

I'm just curious what is meant by this statement. How do you define a function that is "ultimately decreasing"? It'd be great if you could find an example of a function that increases over a certain domain but ultimately decreases, in which the integral test applies.

 

[pasmith]

My textbook states that "For the integral test to apply, it is not necessary that f be always decreasing. What is important is that f be ultimately decreasing."

I'm just curious what is meant by this statement. How do you define a function that is "ultimately decreasing"?

A function f is "ultimately decreasing" if there exists R such that f(x) > f(y) whenever R \leq x < y.

It'd be great if you could find an example of a function that increases over a certain domain but ultimately decreases, in which the integral test applies.

Consider
\sum_{n = 0}^{\infty} \frac{1}{1 + (n - 5)^2}

f(x) = (1 + (x - 5)^2)^{-1} is decreasing for x \geq 5. The fact that f is increasing for 0 \leq x \leq 5 does not affect convergence. We have
<br /> \int_0^{\infty} \frac{1}{1 + (x - 5)^2}\,\mathrm{d}x = <br /> \left[ \arctan(x - 5)\right]_{0}^{\infty} = \frac{\pi}{2} - \arctan(-5) = \frac{\pi}{2} + \arctan(5)
so the series converges.