chapter 8 (sequences and series)

Question 1

seqquence

Answer 1

an ordered list of numbers

Question 2

explicit formula of a sequence

Answer 2

an = f(n)

Question 3

recoursive formula of a sequence

Answer 3

an+1 = f(an)

Question 4

limits of a sequence

Answer 4

if the limit exists then it converges

if the limit does not exist than it diverges

Question 5

infinite series

Answer 5

the sum of the terms in a sequence

Question 6

lim(to inf) (a+b) =

Answer 6

a+b

Question 7

lim(to inf) cA = (c is a real number)

Answer 7

cA

Question 8

lim(to inf) AB

Answer 8

AB

Question 9

increasing

Answer 9

if a(n+1)>an 1,2,3,4,5

Question 10

noindecreasing

Answer 10

if a(n+1) >= an 1,2,2,3,3,4,4,5

Question 11

decreasing

Answer 11

if a(n+1)

Question 12

nonincreasing

Answer 12

if a(n+1)<=an 5,5,4,4,3,3,2,2,1,1

Question 13

monotonic

Answer 13

if it is either nonincreasing or nondecreasing(moves in one direction)

Question 14

bounded

Answer 14

if there is a number M such that |a|n<=Mfir all relevant values of n

Question 15

geometric sequence

Answer 15

have the form r^n or ar^n

Question 16

what must be true for it to be geometric sequence

Answer 16

a!=0 && r!=0 must have a constant R

Question 17

convergence and divergence of a geometric sequence

Answer 17

if abs(r) < 1   = converges 0
if r=1    = converges 1
if r<=-1 ||r>1 divgs

Question 18

bounded monotonic sequence

Answer 18

converges

Question 19

geometric series

Answer 19

in form ar^k a!=0 and r is != 0

Question 20

convergence and divergence of a geometric series

Answer 20

a!=0
r=real number

if abs(r)<1 then a/1-r is what it converges to
if abs(r)>=1 then it diverges

Question 21

telescoping series

Answer 21

get it in teh form where it is subtracting the next term consecutively

partial fraction decomp might be needed

then justify which terms are deleted by listing them
usually just the first and the least
take the limit

Question 22

divergence test

Answer 22

take the limit of the series
if the limit is =0 then the test is inconclusive
if the limit is !=0 then the test prooves divergence

Question 23

harmonic series

Answer 23

1/k diverges even though the terms approach 0

Question 24

integral test

Answer 24

f is continuous
f is positive
f is decreasing(derive and see if negative)
take the integral approaching infinity if it converges then the series converges and if it diverges than the series diverges

Question 25

p series1/k^p

Answer 25

converges for p>1 | diverges for p<=1

Question 26

ratio test

Answer 26

f is continuous f is positive r= (f(a(n+1))/f(a)) if 0<=r<1 series converges if r>1 series diverges if r = 1 test inconclusive

Question 27

root test

Answer 27

f is continuous f is positive p = ksqrt(ak) if 0<=p<1 series converges if p>1 series diverges if p = 1 test inconclusive

Question 28

comparison test

Answer 28

given ak choose a bk that makes things simple both must be cont and pos if bk convgs and ak bk then ak diverges

Question 29

limit comparison test

Answer 29

given ak choose a bk that makes things simple both must be cont and pos lim(to inf) (ak/bk) = L if 0

Question 30

steps for determining a test

Answer 30

1. try divergence test 2. check to see if it is special ie geometric or p 3. see if you can integrate it( if so try integral test0 4. if it involves a !, k^k or a^k the ratio test is good, if ^k is present than the root test may be good 5. if k is a rational function of k use either the comparison test or the limit comparison test

Question 31

alternating series tests

Answer 31

ak is severything but the (-1)^k ak>0 ak>ak+1 = ak converges lim ak = 0

Question 32

absolute convergence and divergence

Answer 32

both the absolute value and the actual series converge or diverge

Question 33

conditional convergence and divergance

Answer 33

either of the abs or the regular either diverge or converge

Question 34

find an n such that the remainder is rem

Answer 34

find ak(like in alt series) set it as ak+1

Question 35

estimate alt series

Answer 35

find ak+1 plug that in with the given error ak+1