Concept preview

What you will be able to do

1. 01State the integral test: if $a_n = f(n)$ for a function $f$ that is positive, decreasing, and continuous on $[1,\infty)$, then $\sum a_n$ and $\int_1^{\infty} f(x)\,dx$ both converge or both diverge.
2. 02Verify the three hypotheses (positive, decreasing, continuous) before applying the test, and evaluate the relevant improper integral as a limit.
3. 03Use the integral test to classify the $p$-series $\sum 1/n^p$ (converges for $p>1$, diverges for $p\le 1$), while understanding that the integral value does not equal the sum.

Suppose the terms of a positive series come from a function, $a_n = f(n)$, where $f$ is positive, decreasing, and continuous for $x \ge 1$. Draw the curve $y = f(x)$ and, on each interval $[n, n+1]$, a rectangle of width $1$ and height $f(n)$. Because $f$ is decreasing, that rectangle sits ABOVE the curve on its interval, so $f(n) \ge \int_n^{n+1} f(x)\,dx$. Alternatively, using height $f(n+1)$, the rectangle sits BELOW the curve. Summing these rectangle-versus-area comparisons across all $n$ traps the series between two copies of the integral.