Practice question · Fill in the blanks
Complete the statement about decay and convergence.
For an improper integral over an infinite interval, the integrand tending to zero is ______ for convergence.
Word bank: necessary but not sufficient · sufficient but not necessary · both necessary and sufficient · neither necessary nor sufficient
Hints
- Find a function that tends to zero yet still encloses infinite area.
- Compare 1/x with 1 over x squared.
Show the answer
For an improper integral over an infinite interval, the integrand tending to zero is necessary but not sufficient for convergence.
Why
1/x tends to zero, yet its integral from 1 to infinity diverges, so decay is not sufficient. It is necessary in the sense that a non-decaying tail cannot give a finite total. What matters is how fast the decay happens.
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.
More questions on Improper Integrals
- ∫₀^∞ e^(−x²) dx converges to √π/2, a finite answer involving π, for a function with no elementary…
- ∫₁^∞ (1/x²) dx converges to 1 while ∫₁^∞ (1/x) dx diverges, although both integrands shrink to zero. Why does…
- Select every integral that is improper.
- If f(x) tends to 0 as x tends to infinity, then the integral of f from 1 to infinity must converge.