APBC calc tests for convergence and divergence

Nth-term test for DIVERGENCE

always do first

diverge if limit ≠ 0

Geometric series test

if r < 1 converge

if r >= 1 diverge

find S = a/1-r

Ratio test DO ENDPOINTS!!!

find An+1

limit ABOLUTE VALUE then DIVIDE An+1 by An

if limit < 1, series converges

if limit > 1, series diverges

if limit = 1, test is inconclusive

write everything out, declare terms

Integral test

f(x) MUST BE

POSITIVE, DECREASING, CONTINUOUS

find the integral of bound

if the limit of integral ≠ infinity converges

if the limit of integral = infinity diverges

P-series test

for pure 1/n

p > 1 converges

p ≤ 1 diverges

Direct Comparison Test

POSITIVE

mutated series, but not as much

SIMPLIFY, find CONVERGENCE

big converge small converge >

small diverge big diverge <

Limit Comparison Test

POSITIVE

Limit, Longer Test + Terms

SIMPLIFY, find CONVERGENCE

LIMIT of NORMAL / SIMPLIFIED =

if it approaches FINITE number, both converge or diverge

Alternating Series Test for CONVERGENCE

MUST BE:

-alternating

-decrease in magnitude (compare An > An+1)

-have limit of 0

Then, REMOVE the alternator, take the limit, and use nth term test

If conditions match, series converge

Absolute and Conditional Convergence

Check convergence for abs value and normal

-if abs value of An converge, absolute convergence

-if An converge but abs value of An don’t, conditional convergence

integral tan inverse formula

1/a tan-1(u/a) + c

integral sin inverse

sin -1 (u/a) +c

Alternating Series Remainder

expand

must write: since series is

• decreasing in magnitude

• alternating

• limit of zero

  Sn - error <= Sn <= Sn + error

error is the absolute value of the next term

show that error < fraction number

Alternating series ERROR

WRITE CONDITIONS FOR AST

alternating, decrease, limit of 0

find term that is less than error

state that S - abs error <= Sum <= S + abs error

declare all terms