Mathematics
Convergence Tests for Infinite Series: Ratio and Root Tests
Quick fact
The ratio test can instantly tell you that the series 1 + 1/2! + 1/3! + ... (the sum of reciprocal factorials) converges to e, while the root test can handle series like Σ (1/n^n) with ease—both tests reveal convergence by comparing the terms to those of a geometric series.
Why this is interesting
Have you ever wondered how mathematicians decide whether an infinite sum like 1 + 1/2 + 1/4 + ... has a finite value? Some series clearly diverge, but for many, the answer is not so obvious—so we need clever tests to determine their fate.