Question 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.
Consider
State and apply geometric convergence for |r| < 1 Apply the p-series test: converge iff p > 1 Compare unknown series to geometric or p-benchmarks Correct myths about harmonic and 'always geometric' convergence
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Σ (1/2)ⁿ exponential decay settles quickly Σ 1/n slow power decay never settles
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