Testing ∫₀∞ e⁻ˣ² dx

Real Analysis · Axiom Academy

EXAMPLE Testing for Convergence Using the comparison test to prove the Gaussian integral converges — without ever evaluating it Determine whether the improper integral converges or diverges . The catch: e^ -x^2 has no elementary antiderivative, so we cannot evaluate it directly — we must establish convergence another way. For , the integrand e^ -x^2 (blue) is squeezed below the larger function e^ -x (red). The shaded gap shows on — the inequality that drives the comparison. Nice work — you proved converges using only a comparison, never an antiderivative. Split, then conquer: when direct integration is impossible, break the interval at a convenient point and handle each piece separately. Finite intervals are free: a function that is continuous and bounded on a closed interval like [0,1] always gives a finite (convergent) integral. Find a comparison: for we used to get — bounding the hard integrand above by an easy one. The Comparison Test: if and converges, then converges too. The exact value: this is the famous Gaussian integral — — but elementary methods can only prove it converges, not evaluate it. Comparison is a workhorse of real analysis: you will reuse this exact split-and-bound strategy to settle the convergence of integrals and series throughout the course.

This is the written version of the interactive lesson above. See the full Real Analysis course.