I know the tests but I freeze on which one to start with, and I end up trying three before something works.
Accepted answer
Work down a fixed checklist instead of picking by feel — the order matters more than knowing extra tests.
1. **nth-term test first, always.** If `aₙ` doesn't approach 0, the series diverges and you're done. (The converse is false: `aₙ → 0` proves nothing.)
2. **Do you recognize it?** Geometric (`Σarⁿ`, converges when `|r| < 1`) and p-series (`Σ1/nᵖ`, converges when `p > 1`) are worth spotting instantly.
3. **Factorials or `n`th powers?** Ratio test — those cancel cleanly.
4. **Looks like a p-series with clutter?** Limit comparison against the dominant term.
5. **Alternating `(−1)ⁿ`?** Alternating series test: terms decreasing and going to 0.
One trap: the ratio test is **inconclusive** when the limit equals 1, which is exactly what happens on p-series. If you get 1, switch tests rather than concluding anything.