The main methods for checking if a series converges or diverges include the comparison test, where if 0 ≤ a_n ≤ b_n for all n and the sum of b_n converges, then the sum of a_n converges, and if the sum of a_n diverges, then the sum of b_n diverges; the limit comparison test, which asserts that if lim(n→∞) (a_n/b_n) = c, where c is a positive finite number, then either both series of a_n and b_n converge or both diverge; the integral test, where if f is a continuous, positive, and decreasing function on [1, ∞) and a_n = f(n), then the series of a_n converges if and only if the integral from 1 to ∞ of f(x) dx converges; the ratio test, which states that if lim(n→∞) (|a_(n+1)/a_n|) = L, the series converges when L < 1, diverges when L > 1, and is inconclusive when L = 1; the root test, which establishes that if lim(n→∞) (|a_n|)^(1/n) = L, the series converges for L < 1, diverges for L > 1, and is inconclusive for L = 1; the alternating series test, which posits that an alternating series ∑(-1)^n a_n converges if a_n is positive, decreasing, and lim(n→∞) a_n = 0; the absolute convergence test, where if the series of |a_n| converges, then the series of a_n also converges; and the p-series test, which determines that the series ∑1/n^p converges if p > 1 and diverges if p ≤ 1, and by applying these methods in various combinations, one can determine whether a given series converges or diverges, depending on the specific characteristics of the series in question.