Practice question · True or false
If f(x) tends to 0 as x tends to infinity, then the integral of f from 1 to infinity must converge.
Hints
- Look for a counterexample among the powers 1/x^p.
- Consider p = 1 specifically.
Show the answer
False
Why
False. 1/x tends to zero yet its integral from 1 to infinity is infinite. Convergence depends on the rate of decay, not on the mere fact of it, this is the pitfall the lesson singles out, and the same one that catches students in the theory of series.
Practise Improper Integrals
The app has 6 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.