chapter 8 (sequences and series) Flashcards

1
Q

seqquence

A

an ordered list of numbers

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2
Q

explicit formula of a sequence

A

an = f(n)

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3
Q

recoursive formula of a sequence

A

an+1 = f(an)

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4
Q

limits of a sequence

A

if the limit exists then it converges

if the limit does not exist than it diverges

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5
Q

infinite series

A

the sum of the terms in a sequence

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6
Q

lim(to inf) (a+b) =

A

a+b

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7
Q

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

A

cA

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8
Q

lim(to inf) AB

A

AB

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9
Q

increasing

A

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

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10
Q

noindecreasing

A

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

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11
Q

decreasing

A

if a(n+1)

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12
Q

nonincreasing

A

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

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13
Q

monotonic

A

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

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14
Q

bounded

A

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

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15
Q

geometric sequence

A

have the form r^n or ar^n

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16
Q

what must be true for it to be geometric sequence

A

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

17
Q

convergence and divergence of a geometric sequence

A
if abs(r) < 1   = converges 0
if r=1    = converges 1
if r<=-1 ||r>1 divgs
18
Q

bounded monotonic sequence

A

converges

19
Q

geometric series

A

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

20
Q

convergence and divergence of a geometric series

A

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
21
Q

telescoping series

A

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

22
Q

divergence test

A

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

23
Q

harmonic series

A

1/k diverges even though the terms approach 0

24
Q

integral test

A

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

25
Q

p series1/k^p

A

converges for p>1

diverges for p<=1

26
Q

ratio test

A

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

27
Q

root test

A

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

28
Q

comparison test

A

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

29
Q

limit comparison test

A

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

30
Q

steps for determining a test

A
  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
31
Q

alternating series tests

A

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

32
Q

absolute convergence and divergence

A

both the absolute value and the actual series converge or diverge

33
Q

conditional convergence and divergance

A

either of the abs or the regular either diverge or converge

34
Q

find an n such that the remainder is rem

A

find ak(like in alt series)

set it as ak+1

35
Q

estimate alt series

A

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