2. Series Convergence Advanced Tester — Theory
Decide whether \(\sum a_n\) converges using classical calculus tests (comparison, limit comparison, integral, etc.).
Many tests can be inconclusive, so the calculator may try a “test chain”.
What convergence means
Define partial sums \(S_N=\sum_{n=n_0}^{N}a_n\). The series converges if \(S_N\) approaches a finite limit.
\[
\sum_{n=n_0}^{\infty} a_n \text{ converges } \Longleftrightarrow
\lim_{N\to\infty} S_N \in \mathbb{R}.
\]
The graph/diagnostics are computed on a finite window. They help intuition, but the selected test(s) decide convergence.
Necessary condition (quick fail)
If \(\sum a_n\) converges, then \(a_n\to 0\).
\[
\sum a_n \text{ convergent } \Longrightarrow \lim_{n\to\infty} a_n = 0.
\]
If \(\lim a_n\neq 0\) or does not exist, the series diverges (often called the \(n\)-th term test).
Reference series you compare to
- Geometric: \(\sum ar^n\) converges iff \(|r|<1\).
- \(p\)-series: \(\sum \frac{1}{n^p}\) converges iff \(p>1\).
- Log-corrected: \(\sum_{n\ge 2}\frac{1}{n(\ln n)^p}\) converges iff \(p>1\).
Comparison test
Best for nonnegative terms \(a_n\ge 0\) (eventually). Choose \(b_n\ge 0\) with known behavior.
\[
0\le a_n \le b_n \text{ for all large }n
\]
- If \(\sum b_n\) converges, then \(\sum a_n\) converges.
- If \(\sum a_n\) diverges, then \(\sum b_n\) diverges.
Limit comparison test (fixed)
For positive \(a_n,b_n\) (eventually), compute
\[
L=\lim_{n\to\infty}\frac{a_n}{b_n}.
\]
- If \(0<L<\infty\), then \(\sum a_n\) and \(\sum b_n\) both converge or both diverge.
- If \(L=0\) and \(\sum b_n\) converges, then \(\sum a_n\) converges.
- If \(L=\infty\) and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.
- Otherwise: inconclusive.
Integral test
If \(a_n=f(n)\) where \(f\) is positive, continuous, and decreasing on \([n_0,\infty)\), then:
\[
\sum_{n=n_0}^{\infty} a_n \text{ converges } \Longleftrightarrow
\int_{n_0}^{\infty} f(x)\,dx \text{ converges.}
\]
Example: \(\sum_{n=2}^{\infty}\frac{1}{n\ln n}\) diverges since
\(\int_2^\infty \frac{dx}{x\ln x}=\ln(\ln x)\big|_2^\infty=\infty\).
Ratio / Root tests (often quick)
\[
\text{Ratio: }L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|,\qquad
\text{Root: }R=\limsup_{n\to\infty}\sqrt[n]{|a_n|}.
\]
- If \(L<1\) (or \(R<1\)): converges absolutely.
- If \(L>1\) (or \(R>1\)): diverges.
- If \(L=1\) (or \(R=1\)): inconclusive.
Alternating series & absolute vs conditional
If \(a_n = (-1)^n b_n\) with \(b_n\ge 0\):
\[
\text{If } b_n \downarrow 0,\ \text{then } \sum (-1)^n b_n \text{ converges.}
\]
Absolute convergence means \(\sum |a_n|\) converges. Otherwise, it may converge conditionally.
