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Math101learn.math101.caIntegral 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.
Fully worked example
Graphical or geometric meaning
Right-endpoint rectangles lie below a positive decreasing curve and left-endpoint rectangles lie above. The same picture yields both the convergence equivalence and the remainder bounds.
Common mistakes and why they fail
Verification and reasonableness checks
- Differentiate the continuous extension to verify decrease.
- Keep integral and series conclusions distinct.
- Check remainder bounds place a numerical approximation in a sensible interval.
Match the sequence to a decreasing function
The integral test needs $a_n=f(n)$ for a function continuous, positive, and decreasing eventually. Then the series and corresponding improper integral share convergence behavior. The starting index changes finite values but not classification. In a convergent case, $\int_{N+1}^\infty f(x)dx\le R_N\le\int_N^\infty f(x)dx$, giving a practical error bound. Do not skip monotonicity merely because $f$ is positive; the rectangle comparison depends on decrease. It is enough to establish these properties beyond some index, since finite initial terms do not matter. When solving for a target accuracy, identify whether the upper or lower bound supplies the needed guarantee and keep the partial-sum index consistent.
Practice
- Use the integral test on $\sum1/n^2$.
- Use it on $\sum1/n$.
- Does the series equal the comparison integral?
Answers and brief solutions
- It converges.
- It diverges.
- Not generally.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What does the integral test conclude about Σ from n=2 to ∞ of 1/[n(ln n)²]?
- The function is positive, continuous, and decreasing for x≥2.
- The integral becomes ∫ from ln2 to ∞ of u⁻²du.
- That integral is finite, so the series converges.
End of lesson
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