Series Tests Geometric Series: Σ ar.-a+ ar + ar' +..tar" + (witha # 0) converges if I < 1 and vrges f 21 Divergence Test: If lima,#0, Σα.dverges. Integral Test: Suppose fis continuous, positive, decreasing function (at least eventually) with 4° f(n). Then, if If(x)ax converges, Σα. converges; if I f(x)ax dverges, Σα. diverges. p-series: Σ +-+-+-+.. converges if p > 1 and diverges if 0 < p 1 The comparison tests must have positive terms (1) Direct Comparison b. and Σ>, converges, then Σ a, converges. If a, If b, s a, and Σ4, diverges, then Σ a, diverges. (2) The Limit Comparison Test: If lim - 1 andan fin, then both Ya and b, behave the same. That is, both series converge or both diverge Alternating Series Test for either form Σ(-1)"'q or Σ(-1)"a, (1) Ignoring the + signs, check that the terms are decreasing. That is, make certain that (2) Check that the terms are heading to zero. That is, make sure that lima-o If both conditions are satisfied, then the alternating series converges. The Ratio Test for Absolute Convergence lflimPatil-L and L < 1 , then the series Σ a, converges absolutely. If L > 1 , then the series diverges. If L = 1 , then the test provides no useful information. The Root Test for Absolute Convergence !flim Vla,-L and L < 1 . then the series Σ a, converges absolutely. If L > 1 , then the series diverges. If L = 1 , then the test provides no useful information.