Comparison, Ratio, and Root tests for absolute convergence

A few useful convergence theorems for real-valued sequences.  The theorems are stated in terms of absolute convergence where possible since that implies convergence.

Comparison Test: If |a_n| ≤ |b_n| for all n, then if Σ|b_n| converges, Σ|b_n| converges and Σ|a_n| ≤ Σ|b_n|. Stated as the contrapositive, if Σ|a_n| diverges then Σ|b_n| diverges.

Ratio Test: Let L = lim |(a_n+1)/a_n| (possibly infinite). If L < 1 then Σ|a_n| converges. If L > 1 then Σa_n diverges. It's anybody's guess if L = 1.

Root Test: Let L = lim sup |a_n|^1/n (possibly infinite). If L < 1 then Σ|a_n| converges. If L > 1 then Σa_n diverges. Again, all bets are off if L = 1.