Calc 2

nth term divergence test

If limit is DNE or not equal to zero, the series diverges.

p-series

if p > 1 series converges
if p <= 1 series diverges

integral test

if integral converges series converges (equal a finite number)
if integral diverges series diverges

limit comparison test

Limit an / bn = L
if L > 0 & bn converge, an converges, vice versa
if L = 0 & bn converge, an converge
if L = infinity & bn diverge, an diverge

direct comparison test

if an <= bn and bn converges, then an converges; if an >= bn and bn diverges, then an diverges.

ratio test

limit | an+1 / an | = L
L < 1, series converges
L > 1, series diverges
L = 1, test inconclusive.

root test

limit | an^(1/n) | = L
L < 1, series converges
L > 1, series diverges
L = 1, test inconclusive.

power series

f(x) + f’(x)(x - c) + f’’(x)/2! (x - c)² + f’’’(x)/3! (x-c)³

integral of maclaurin or taylor

t^m+1 / m+1