Series Convergence/Divergence Review

form of geometric series

a_n = ar^n

geometric series converges

absolute value of r < 1; a_n = a/(1-r)

geometric series diverges

absolute value of r ≥ 1

form of p-series

a_n = 1/(n^p)

p-series converges

p>1

p-series diverges

0<p≤1

form of telescoping series

a_n - b_n

telescoping series converges

subsequent terms cancel previous terms. converges to non-cancelled terms

telescoping series diverges

no general cancellation of terms

form of alternating series

(-1)^n(a_n), (-1)^(n+1)(a_n)

alternating series converges

a_(n+1) < a_n

alternating series diverges

a_n+1 ≥ a_n

ratio test converges

limit of absolute value of (a_(n+1))/(a_n) < 1

ratio test diverges

limit of absolute value of (a_(n+1))/(a_n) > 1

ratio test is inconclusive

limit of absolute value of (a_(n+1))/(a_n) = 1

limit comparison test converges

b_n is convergent and the limit of (a_n)/(b_n) = L (positive and finite)

limit comparison test is divergent

b_n is divergent and the limit of (a_n)/(b_n) = L (positive and finite)

integral test converges

f(n) = a_n, f is continuous and decreasing on [a, infinity), integral from a to infinity of f(x) = L

integral test diverges

the integral from a to infinity of f(x) does not exist

root test converges

limit of the absolute value of (a_n)^(1/n) < 1

root test diverges

limit of the absolute value of (a_n)^(1/n) > 1

root test is inconclusive

limit of the absolute value of (a_n)^(1/n) = 1

direct comparison test converges

a_n < b_n for all n, if b_n converges, then a_n converges

direct comparison test diverges

a_n < b_n for all n, if a_n diverges, then b_n diverges

form of nth term test

limit of a_n

nth term test diverges

limit of a_n ≠ 0

nth term test is inconclusive

limit of a_n = 0