Mathematical Methods in the Physical Sciences · Problem 6.8

The integral test connects infinite series to improper integrals: if you can evaluate $\int_1^{\infty} \frac{x}{x^2+4}\,dx$, its behavior tells you whether the series converges. The key insight is recognizing that the denominator's derivative appears in the numerator—a classic setup for a $u$-substitution that transforms this into a logarithmic integral, which grows without bound.

The integral test connects infinite series to improper integrals: if you can evaluate $\int_1^{\infty} \frac{x}{x^2+4}\,dx$, its behavior tells you whether the series converges. The key insight is recognizing that the denominator's derivative appears in the numerator—a classic setup for a $u$-substitution that transforms this into a logarithmic integral, which grows without bound.