Calc 2 Series Convergence Tests

geometric series test

A geometric series ∑ar^n-1 is convergent if |r|<1 and divergent if |r|>1

divergence test

if lim aₙ≠0 or DNE, the series ∑aₙ is divergent. If lim aₙ=0, the test is inconclusive

integral test

let ∑aₙ be a series with positive terms aₙ≥0 and aₙ=f(x) where f(x) is continuous decreasing and positive on [1,∞]. If ∫₁^∞f(x)dx is convergent, ∑aₙ is convergent, and if ∫₁^∞f(x)dx is divergent, ∑aₙ is divergent

use if you see ln

p-series test

Σ1/n^p is convergent if p>1 and divergent if p≤1

direct comparison test

∑aₙ and ∑bₙ are both series with positive terms. If aₙ≤bₙ and Σbₙ is convergent, Σaₙ is convergent. If Σaₙ is divergent, ∑bₙ is divergent

limit comparison test

∑aₙ and ∑bₙ are both series with positive terms. If lim_n→∞ aₙ/bₙ = c where c is a positive finite number, then either both series converge or both series diverge

alternating series test

∑aₙ is an alternating series if aₙ=(-1)ⁿbₙ or aₙ=(-1)^n-1bₙ. where bₙ is positive. If bₙ₊₁<bₙ and lim_n→∞ bₙ=0, then the alternating series is convergent. If any of these conditions is broken, the test is inconclusive

absolute convergence test

if Σ|aₙ| is convergent, Σaₙ is absolutely convergent and thus convergent. If Σaₙ is convergent but Σ|aₙ| is divergent, Σaₙ is conditionally convergent

use if series has negative terms but is not alternating (sin, cos)

ratio test

if lim_n→∞|aₙ₊₁/aₙ| < 1, Σaₙ is absolutely convergent

if lim_n→∞|aₙ₊₁/aₙ| > 1, Σaₙ is divergent

if lim_n→∞|aₙ₊₁/aₙ| = 1, the test is inconclusive

use if there are factorials

root test

if lim_n→∞ ⁿ√|aₙ| < 1, Σaₙ is absolutely convergent

if lim_n→∞ ⁿ√|aₙ| >1, Σaₙ is divergent

if lim_n→∞ ⁿ√|aₙ| = 1, the test is inconclusive

use if there are powers involving n and no factorials