7.8 Improper Integrals
I = - integral from 0 to infinity of tan-1(x) / (1+x^2) dx
Answer:
A(t) = integral from 0 to t of tan-1(x) / (1+x^2) dx
u = tan-1(x) -> du = 1/(1+x^2) dx
A(t) = integral from 0 to tan-1(x) of u du = 1/2 u^2 from 0 to tan-1(t)
=

Review
Theorem 1 Comparison Test
given cfw_an and cfw_bn positive sequences
If an <= bn for n>= K and sum bn converges then sum an converges
If an >= bn for n>=K and sum bn diverges then sum an diverges
Theorem 2 Limit Comparison Test
Given cfw_an and cf

Taylor and Mackurin Series
Taylor Series:
Informal way
f(x) = sum from n=0 to infinity of f^(n)(a)/n! * (x-a)^n
Mackuran:
f(x) = sum from n=0 to infinity of f^(n)(a)/n! * x^n
Partial Sum:
Tn(x) = sum from i=0 to n of f^(i)(a)/i! * (x-a)^i
(n-th de

Review
Theorem 1 Comparison Test
given cfw_an and cfw_bn positive sequences
If an <= bn for n>= K and sum bn converges then sum an converges
If an >= bn for n>=K and sum bn diverges then sum an diverges
Theorem 2 Limit Comparison Test
Given cfw_an and cfw

Review:
given cfw_an the series is devined as the sum from n=1 to infinity of an = a1 + a2 + an
Partial Sum:
sn = sum from i=1 to n of ai = a1 + a2 + an
Definition
if cfw_Sn converges then sum from n=1 to infinity of an converges
if cfw_Sn diverges

11.5 Alternating series
Definition
An alternating series is denoted by sum from =1 to infinity of (-1)^n * an = -a1 + a2 - a3 + a4 etc
Alternating Test
If an > 0 and an+1 <= an for n>= k
limit as n-> infinity of an = 0
then the sum of (-1)^n * an co

11.6 Absolute convergence, the ratio and root test
Definition: A series sum from n=1 to infiity of an is absolutely convergent (AC) if sum from n=1 to infinity of |an| converges
Definition: A series n=1 to infinity of an is called "conditional convergent"

Improper Integrals:
A comparison test:
Given f and g continuous with f(x) >= g(x) >= 0 for all x > a
If integral from a to infinity of f(x) dx converges then integral from a to infinity g(x) also converges
If integral from a to infinity of g(x) dx div

sum 4^(n-1)/3^n-2 = 4^(n_1)/3^n = (4/3)^n
Review: Gvien a secuqnces cfw_an
sum from n=1 to infinity of an = a1 + a2 + an
Partial Sum
Sn = sum from i=1 to n of ai = a1+a2+an
If cfw_Sn converges then
sum from n=1 to infinity of an = limit as n-> infi

11.1 Sequences
A sequence is a list of numbers
cfw_an] has limit L if an->L as n-> infinity
limit as n->infinity of an = L converges
Theorem 1
Given f IR -> IR such that
f(n)=an for all in IN
limit as x->infinity of f(x) = L
limit as n-> infinity of

Review THeorem (Integral Test) Given f continuous positive and decreasong for f(n) an
a. If integral from 1 to infinity of f(x) dx converges then the sum from n=1 to infinity of an converges
b. If the integral from 1 to infinity of f(x) dx diverges then

Review: Power Series
Definition: A power series is a series
sum from n=0 to infinity of Cn * x^n
Cn given constants, x variables
+ a series of form
sum from n=0 to infinity of Cn (x-a)^n
is called a "power series" in (x-a)
Theorem 1: Given a power se