Series rules

Geometric series formula \sum_{n=0}^{\infty} a\,r^n = \frac{a}{1 - r}, \quad \lvert r\rvert < 1 The geometric series formula: a + ar + ar² + ... = a/(1 − r) when |r| < 1. Partial sums, the convergence condition and worked examples. nth-term test \lim_{n\to\infty} a_n \ne 0 \;\Longrightarrow\; \sum a_n \text{ diverges} The nth-term test: if the terms of a series do not approach 0, the series diverges. The statement, why it cannot prove convergence, and examples. Ratio test L = \lim_{n\to\infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert The ratio test: compute L = lim |a(n+1)/a(n)|. If L < 1 the series converges absolutely, if L > 1 it diverges, and L = 1 is inconclusive. Alternating series test b_n \ge b_{n+1} > 0,\ \lim_{n\to\infty} b_n = 0 \;\Longrightarrow\; \sum (-1)^{n+1} b_n \text{ converges} The alternating series test (Leibniz test): if bₙ decreases to 0, then the series of (−1)ⁿbₙ converges. Conditions, the error bound and examples.