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Judging a series by a known neighbor

• If a mystery series looks like a series you already understand, compare them.
• The comparison tests 比较判别法 measure an unknown positive series against a known one (usually a p-series or geometric series).
• Bigger than something that diverges → diverges. Smaller than something that converges → converges.
• Two flavors: a direct comparison and a limit comparison.

Direct Comparison Test

• For positive terms with $0\le a_n\le b_n$:
• If $\sum b_n$ converges, then the smaller $\sum a_n$ converges too.
• If $\sum a_n$ diverges, then the bigger $\sum b_n$ diverges too.
• Squeeze from the right side: bounded above by a convergent series, or below by a divergent one.

Each term is smaller than a term of a convergent geometric series, so the series converges

Limit Comparison Test

• Sometimes inequalities are awkward; instead compare growth rates.
• Take $\displaystyle L=\lim_{n\to\infty}\dfrac{a_n}{b_n}$. If $L$ is a finite positive number ($0<L<\infty$), then $\sum a_n$ and $\sum b_n$ share the same fate.
• Pick $b_n$ from the dominant terms of $a_n$ (e.g. compare $\tfrac{n}{n^3+1}$ with $\tfrac1{n^2}$).
• The limit being positive and finite means they behave alike.

Choosing the comparison series

• Look at the dominant behavior of $a_n$ for large $n$ — the highest powers on top and bottom.
• $\dfrac{3n^2+1}{n^4+5}$ behaves like $\dfrac{3n^2}{n^4}=\dfrac{3}{n^2}$ — compare with the p-series $\tfrac1{n^2}$.
• Then either bound it directly or take the limit ratio.
• The right comparison series turns a scary series into a known one.