Owl: Some series are so well-behaved we treat them as measuring sticks. Geometric series Σ arⁿ scale by a constant ratio each time. p-series Σ 1/n^p decay like a power of n.
Why does Σ (1/2)ⁿ feel so different from Σ 1/n? One shrinks exponentially; the other only polynomially — and that gap decides whether the running total settles.
We'll nail the |r| < 1 rule, the p-test, and how to compare an unknown series to these benchmarks. Along the way we'll bury two stubborn myths: that the harmonic series converges, and that every geometric series converges.