Improper Integral Guide
Do ∫ k/xᵖ dx over infinite or singular intervals converge? Find out — and get the value.
These are p-integrals: the exponent p alone decides convergence. Try p = 2 vs p = 0.5.
① Answer
② Steps
③ Why it works
An improper integral is a limit in disguise: ∫a∞ really means limb→∞ ∫ab. For k/xᵖ the antiderivative is a power (or ln when p = 1), and whether the far end vanishes or explodes depends only on the sign of 1 − p. That is the whole p-test: the tail ∫∞ converges iff p > 1; the singularity ∫0 converges iff p < 1.