[AP Calc] Unit 9: Series and Sequences

partial sum

geometric series

|r|<1 → convergent

|r|≥1 → divergent

harmonic series

always divergent

test for divergence

if the lim(a) DNE or lim(a)≠0 → the series is divergent

integral test

only used if the function is continuous, positive, and decreasing

take the integral, then the limit

p-series

p>1; convergent

p≤1; divergent

remainder estimate for the integral test

where is f continuous, positive, and decreasing for x≥n and Σa is convergent

comparison test

if Σa and Σb are series with positive terms

limit comparison test

where c is positive and finite and Σa and Σb are series with positive terms

alternating series

a series whose terms alternate from positive to negative

alternating series test

alternating series error bound theorem

absolutely convergent

a series, Σa, is ______ if Σ|a| is convergent

conditionally convergent

a series, Σa, is ______ if Σ|a| is divergent

ratio test

root test