Convergence tests
Most series have no formula for their partial sums. How can we still decide whether adding infinitely many terms settles on a number?
Where you are 6 inside
What this area is about
A series \sum a_n converges when its partial sums approach a number. Few series besides geometric ones have a formula for those sums. Convergence tests answer the question from the shape of the terms alone, each one only for series that meet its hypotheses.
How these fit together
Tests for positive terms. The integral test and p series compares a series with \int_m^{\infty} f(x)\,dx when f is continuous, positive and decreasing on [m, \infty), and shows \displaystyle \sum \frac{1}{n^p} converges exactly when p > 1. The direct and limit comparison tests judge a series of nonnegative terms by a known series it sits below or above, or stays in a fixed positive ratio with.
Signs that change. Alternating series and their error bound shows that \sum (-1)^{n+1} b_n converges when b_n decreases to 0, with error at most the first term left out. Absolute and conditional convergence asks whether \sum \lvert a_n \rvert also converges.
Ratio and root. The ratio and root tests settle series with factorials and powers when \displaystyle L = \lim \left\lvert \frac{a_{n+1}}{a_n} \right\rvert is not 1, and say nothing when L = 1.
Strategy. Choosing a convergence test reads the shape of a term to pick the test that will work.
A path through it
Start with the integral test, since its p series are the usual benchmarks for comparison. Comparison and alternating series come before absolute and conditional convergence, which comes before the ratio and root tests. Choosing a test comes last.
Next comes power series and the interval of convergence.