Calc Week 3

Question 1

geometric series test

Answer 1

converges if |r| <1
converge to 1/ 1-r

s sub n = 1 - r^n+1/1-r

Question 2

integral test

Answer 2

let {a sub n} be a sequence of non negative real numbers and let f(x) be a positive, decreasing continuous function defined on [1,inf) such that f(n) = a sub n for every n = 1,2,3…. then
sigma from n =1 to inf of a sub n converges if and only if integral from 1 to inf of f(x) dx converges

Question 3

p-series

Answer 3

a series of the form
sigma from n=1 to inf of 1/n^p when p> 0

general p-series
sigma n=1 to inf of 1/(an+b)^p where a and b are fixed real numbers and p>0

Question 4

p-series test

Answer 4

a general p-series sigma from n=1 to inf of 1/(an+b)^p where an+b !=0 for all n converges if and only if p>1

Question 5

two strategies for evaluating series

Answer 5

1. write out a bunch of terms of {s sub n} and find a formula for it, and take the limit as n approaches infinity
2. check whether the terms of the series converge to 0, if not, you’re done(nth term test)

Question 6

direct comparison test

Answer 6

suppose {a sub n} and {b sub n) satisfy 0<= a sub n <= b sub n for all n then

1. if the sigma from n=1 to inf of b sub n converges, then series a sub n converges
2. if the sigma from n=1 to inf of a sub n diverges, then the series b sub n diverges

Question 7

integral convergence from 1 to inf of 1/x^P

Answer 7

diverges if p <= 1

1/p-1 if p >1

Question 8

momma bear condition

Answer 8

lim n-> inf of a sub n/b sub n = 0

{a sub n} converges at a faster rate as n-> inf than {b sub n}
- {a sub n} is asymptotically smaller than {b sub n}

Question 9

papa bear condition

Answer 9

lim n-> inf of a sub n/b sub n = +inf

terms of {a sub n} decay to zero at a much slower rate than the terms of {b sub n}
- {a sub n} is asymptotically larger than {b sub n}

Question 10

comparable

Answer 10

lim n-> inf of a sub n/b sub n = L
0<L<inf

• each sequence converges to zero, neither too quickly nor too slowly relative to the other
• {a sub n} and {b sub n} are comparable

Question 11

limit comparison test

Answer 11

let {a sub n} and {b sub n} be sequences such that a sub n>= 0 and b sub n > 0 for all n then

1. if lim n->inf a sub n/b sub n = L with 0<L<inf then
   series of a sub n converges if and only if series b sub n converges
2. if lim n-> inf a sub n/b sub n = 0 and the series b sub n< inf then series a sub n < inf
3. if lim->inf a sub n/b sub n = +inf and series b sub n =+inf then series a sub n = +inf

Question 12

how to show comparability

Answer 12

lim n -> inf a sub n/b sub n = L

Question 13

solving with limit comparison test

Answer 13

find a b sub n that is similar to a sub n that you know converges or diverges

take lim n-> inf of a sub n/b sub n

check if they are comparable

use theorem to explain

Question 14

lim n-> 0 of sinx/x

Answer 14

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