Series and Sequences

Divergence Test meaning

diverges when limit as n approaches infinity of a sub n is not 0

Divergence Test series

a sub n

Geometric Test Series

ar^n

Geometric Test meaning

converges when abs value of r <1 then ; diverges when abs value of r equal or >1 ;

sum of geometric test

infinite = (a(1-r^n)/1-r); finite = a/(1-r)

p- test series

1/n^p

p-test meaning

converges when p>1; diverges when p equal to or <1

Alternating Test Series

(-1)^n+1 a sub n

Alternating Test Meaning

Converges when terms are decreasing and limit as n approaches infinity is 0 or abs value of the series is convergent; cannot show divergence

Remainder of AST

abs value of Rn equal to or <a sub n+1

Integral Test Series and conditions

a sub n; when a sub n = f(n) is pos or 0 , continuous, and decreasing

integral test meaning

converges when integral of f(x)dx from 1 to infinity converges; diverges when the integral of f(x)dx from 1 to infinity diverges

remainder of integral test

0 < R < integral from N to infinity f(x)dx

Ratio Test Meaning

converges limit as n approaches infinity abs value (a n+1)/a sub n is less than 1; diverges greater than 1; inconclusive when = 1

Root Test Meaning

converges when limit as n approaches infinity of root a sub n <1; diverges when >1; inconclusive when =1

Direct Comparison Condition

a sub n, b aub n are positive

Direct Comparison converges when:

a sub n < or equal to b sub n and series b sub n is abs convergent

Direct Comparison diverges when:

b sub n is < or equal to a sub n and series b sub n diverges

Limit Comparison Test Condition

a sub n, b sub n are positive

Limit Comparison Test Converges when:

limit as n approaches infinity of a sub n/ b sub n = L is > 0 and series b sub n converges

Limit Comparison Test Diverges when:

limit as n approaches infinity of a sub n/b sub n = L is > 0 and the series b sub n diverges

Telescoping Test Series

a sub n+1 - a sub n

Telescoping Test Meaning

converges limit as n approaches infinity of a sub n = L; cannot prove divergence