AP Calculus Convergence Tests

Which test should you always start with?

nth term test

lim n→∞ an=0

nth term test is inconclusive

lim n→∞ an doesn’t =0

inconclusive, use another test

1/n^p

p >1

series converges by p-series

1/n^p

p≤ 1

series diverges by p-series

∑a(r)ⁿ

|r|

series converges by geometric series

∑a(r)ⁿ

|r|≥1

series diverges by geometric series

which test should you use when you see ∑(an)ⁿ

root test

lim n→∞. ⁿ√|an| < 1

series converges by root test

lim n→∞. ⁿ√|an| > 1

series diverges by root test

which test should use you when you see factorials and exponentials

ratio test

lim n→∞ |an+1/an| < 1

series converges by ratio test

lim n→∞ |an+1/an| >1

series diverges by ratio test

when you see (-1)ⁿ

use alternating series test, if terms are dec in abs value towards 0

lim n→∞ |an| = 0

series converges by alt srs test

lim n→∞ |an| doesn’t = 0

doesn’t converge by alt srs, use nth term test to prove divergence

If you see a series where there would be a u-sub, use

integral test if the series is positive, cont, and dec

lim b→∞ ∫ from b to ∞. = #

series converges by integral test

lim b→∞ ∫ from b to ∞. = ∞

series diverges by integral test

If a series looks < or > a similar series, use

direct comparison test

If your original series is < the comparison, the series converges

< converging

converges

If your original series is > the comparison, the series diverges

> diverging

diverges

If you see n^a +#/n^b +# , sin (1/n), or tan (1/n), you should use

limit comparison test

lim |an/bn|

n→∞

if the lim is positive, the functions match

if you have studied all the way to this flashcard.

you must be really smart. go take a lil break. i’m proud of you. and i don’t understand most of what is on here. if you do, go take a nap. if you don’t, go take a break and come back to it later :)