Calc Series tests

n-th term test

lim_n→∞ a_n≠0, diverges. lim_n→∞ a_n=0, inconclusive

Geometric series test

∑arⁿ, |r|<1 converges, and 0>|r|≥1 diverges

p-series test

∑1/n^p, p>1 converges, p_<1 diverges

Alternating series test

∑(-1)ⁿb_n, b_n+1≤b_n & lim_n→∞=0 converges

ratio test

L=lim_n→∞|a_n+1/a_n|, L<1 converges, L>1 diverges, L=1 inconclusive

Direct comparison test

0≤a_n≤b_n and ∑b_n converges, then ∑a_n converges

0≤b_n≤a_n and ∑b_n diverges, then ∑a_n diverges

Limit comparison test

L=lim_n→∞a_n/b_n, 0<L<∞ both converge or diverge

Integral test

If f(x) is positive, decreasing, and continuous on [1,∞), then ∑a_n is like ∫₁^∞f(x)dx