Math101 · Independent Ontario learning library
Math101Integral Test
A rigorous, example-driven guide to integral test, including hypotheses, method choice, verification, and practice.
The central idea
If $f$ is positive, continuous, and decreasing on $[N,\infty)$ and $a_n=f(n)$, then $\sum_{n=N}^{\infty}a_n$ and $\int_N^\infty f(x)dx$ either both converge or both diverge. When convergent, the remainder obeys $\int_{n+1}^\infty f\le R_n\le\int_n^\infty f$.
Definitions, hypotheses, and notation
The starting point can be moved forward until the hypotheses hold; finitely many early terms do not affect convergence. The integral need not be easy enough to compute exactly—comparison may still decide its convergence. For remainder bounds, the lower and upper integrals differ by one unit shift because rectangles start after the last included term.
The test is especially natural for logarithmic factors and $p$-series. For factorial or rapidly oscillating terms, ratio, root, or alternating tests usually match structure better. Method selection should minimize hypotheses that are awkward to prove.
Conceptual meaning
For a decreasing positive graph, unit-width rectangles built from endpoint heights bracket the area under the curve. Thus finiteness of the continuous tail and finiteness of the discrete tail are equivalent.
A dependable method and decision rule
- Choose a continuous extension $f(x)$ with $f(n)=a_n$.
- Verify positivity, continuity, and eventual decrease.
- Write the corresponding improper integral as a limit.
- Evaluate its convergence, then transfer the conclusion to the series.
- Use the two remainder integrals if an approximation error is requested.
