final

Question 1

r
what is an infinite series, the general term, and when a series converges

Answer 1

sequence an subset of R. Formal sum a1+a2+a3….+an +…… is i finite series, an is general term, Sn = a1+….+an is nth partial sum

series converges iff lim as n->inf Sn exists!

Question 2

what is a geometric series what is GS test

Answer 2

let a, r in R, a ≠ 0. series in a + ar + ar^2 + … + ar^n + …. is a GS, r is ratio

GS Test: If |r| < 1 the. GS converges with sum a (first non-zero term) / 1-r
If |r| >= 1, diverges

Question 3

properties of convergent series

Answer 3

if an bn both converge to a, b
- an + bn comb to a + b
- can converge to ca
- LIM N->INF an = 0 NOT THE OPPOSITE.

Question 4

what is divergence test and what is proof of vanishing condition? (how to find an in general)

Answer 4

div test: given an, if lim n->inf an ≠ 0 THEN an diverges (only tests for diverge)

proof: lim n-> inf an = lim (an + 0)
= lim (an + a1 +….+an-1 - (a1+a2+….+an-1)
= lim Sn - Sn-1
= S - S = 0

Question 5

what is integral test?

Answer 5

If f is positive, continuous, and DECREASING on interval [1,inf) s.t. f(n) = an for all n in N,

an conv iff improper integral of f(x) dx to inf conv and vice versa for diverges

Question 6

what is p-series?

Answer 6

1/n^p

conv if p>1, div if p<=1

Question 7

direct comparison test CT

Answer 7

If 0 <= an <= bn, for all n in N, bn conv, then an conv

if 0<=cn<=an for all n in N, cn diverges, then an diverges

Question 8

alternating series test

Answer 8

conv only

given series (-1)^n+ibn, bn>0 (important)

if bn >= bn+1 for all n in N
and lim n-> bn = 0 (check first)
then CONV.

Question 9

all series: if an bn div, an + bn = div

Answer 9

FALSE. consider 1,-1, 1-1= 0 which conv

Question 10

series: an conv, bn div, anbn div?

Answer 10

FALSE

an is 0, bn is 1, 0 conv

Question 11

an conv, 1/an div

Answer 11

TRUE, via div test as lim an=0 by defn so 1/0 is inf

Question 12

what does it mean to AC and Cc

Answer 12

let an be a series, an ABSOLUTELY CONVERGES if |an| conv. (and an conv)
CONDITIONALLY CONVERGES if |an| DIV and an CONV

Question 13

ratio test

Answer 13

conv or div, given an an ≠ 0, for any n in N

define L = lim n->inf |an+1/an| IN [0,inf) U {inf}
if L < 1 -> an AC
id L > 1 -> an DIV
if L = 1 -> PLEASE USE ANITHWR TEST DOESNTNWORK

Question 14

what is a power series

Answer 14

infinite summation cn(x-a)^n is a POWER SERIES centered at a
cn(x-a)^n is general term
cn = nth term coefficient
a = center

Question 15

taylor series and mclaurin series

Answer 15

f is a fcn who is infinitely différentiable at point a (x=a). PS with cn = f^(n) (a) /n! [nth derivative of f] IS A TAYLOR SERIES

IF a=0, WE HAVE MCLAURIN SERIES

ex: x^n/n! is mclaurin series of y = f(x) = e^x as nth derivative is e^x and e^(0) = 1

Question 16

a radius and interval of convergence?

Answer 16

largest value R in R+ U {inf} st PS AC for |x-a|<R and DIV for |x-a| > R

set I = {xinR | PS converges} is INTERVAL OF CO NBC ERGENCE

Question 17

comparison theorem for integrals c

Answer 17

0<=f(x)<=g(x), for all x in interval, if improper integral g(x) converges, improperly intergral f(x) converges

if f;x) diverges on interval, g(x) diverges on interval