AP Calculus BC / Unit 10: Infinite Sequences and Series
10.4 Integral Test for Convergence
Notes
If $f$ is positive, continuous and decreasing with $f(n)=a_n$, then $\displaystyle \sum a_n$ and $\displaystyle \int_1^\infty f(x)\,dx$ both converge or both diverge.
Practice
$\displaystyle \sum\frac{1}{n^2}$ converges by the integral test because:
The integral is 1, so the series converges.
The integral test value and the series sum are:
The test tells convergence, not the sum.
$\displaystyle \sum\frac{1}{n\ln n}$ (from $n=2$):
Substitute $u=\ln x$.
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