Convergence tests

Check the terms first

If the terms do not go to 0, the series diverges. If they do, that proves nothing: 1/n goes to 0 and its series still diverges.

A geometric series Σ rⁿ converges when |r| < 1. A p-series Σ 1/nᵖ converges when p > 1. Many problems are one of these in disguise.

For factorials and powers like n! or 3ⁿ: find the limit of |aₙ₊₁/aₙ|. Below 1 converges, above 1 diverges, exactly 1 tells you nothing.

Compare a messy series with a famous one it behaves like. If the signs alternate and the sizes shrink steadily to 0, it converges.