Owl: Most series refuse to hand you a closed form for the partial sums. That's fine — you don't always need the exact total. You need a yes/no: does the running total settle?
Convergence tests are a detective kit. Comparison, ratio, root, and integral each fit a different fingerprint: factorials and exponentials love the ratio test; powers of n invite integral or comparison to a p-series.
The art is choosing a test, checking its hypotheses, then concluding. Missteps include treating "ratio test inconclusive" as "diverges," and comparing with the inequality pointing the wrong way.
We'll match fingerprints to tools and practice clean conclusions.