Calc Week 4

Question 1

Ratio test

Answer 1

let {a sub n} be a positive sequence with
lim n-> inf a sub n+1/ a sub n = L then

1. if L<1, then the series of a sub n converges
2. if L > 1, then the series of a sub n diverges
3. if L=1 , then the test is inconclusive

Question 2

Root test

Answer 2

let {a sub n} be a positive sequence with
L = lim n-> inf of (a sub n) ^ 1/n
then

1. if L<1. the series of a sub n converges
2. if L> 1, the series of a sub n diverges
3. if L=1, the test is inconclusive

Question 3

alternating series

Answer 3

let {a sub n} be a positive sequence
a series in the form of

sigma from n=1 to inf of (-1)^n * (a sub n)
or
sigma from n=1 to inf of
(-1)^n+1 * (a sub n)

Question 4

alternating series test

Answer 4

let {a sub n} be a positive sequence, monotonically decreasing sequence such that
lim n-> inf a sub n = 0
then
(-1)^n * (a sub n)
and
sigma from n=1 to inf of
(-1)^n+1 * (a sub n)
both converge

Question 5

converging with absolutes definition

Answer 5

let sigma from n=1 to inf of a sub n be any series
a. if the series of |a sub n| converges, the series absolutely converges
b. if the series of a sub n converges but the series of |a sub n| diverges, the series converges conditionally

Question 6

absolute convergence theorem

Answer 6

suppose sigma from n=1 to inf of |a sub n| converges, then
1. series of a sub n converges
2. the value of the series of a sub n does not depend on the order of the terms