Revision sheet

Convergence

• Converges: the partial sums have a limit.
• $a_n \not\to 0$: diverges. $a_n \to 0$: test another way.

Watch out: the harmonic series diverges.

Integral test, p-series

• Positive decreasing terms: compare with $\int_1^{\infty} f$.
• $\sum 1/n^p$: converges iff $p > 1$.

Watch out: $p = 1$ diverges.

Comparison tests

• Below convergent: converges. Above divergent: diverges.
• Same leading behavior: limit comparison.

Watch out: below a divergent series tells you nothing.

Alternating series

• $b_n$ decreasing to 0: converges.
• Error after $N$ terms $\le b_{N+1}$.
• Absolute vs conditional: test $\sum |a_n|$.

Watch out: the bound uses the first omitted term.

Ratio test

$L = \lim |a_{n+1}/a_n|$: $L < 1$ converges, $L > 1$ diverges, $L = 1$ unknown.

Watch out: use it for factorials and powers; not for p-series.

Choosing a test

nth-term → known series → alternating → ratio → comparison → integral.

Watch out: a divergent absolute series does not make an alternating series diverge.