chapter 8 (sequences and series) Flashcards
seqquence
an ordered list of numbers
explicit formula of a sequence
an = f(n)
recoursive formula of a sequence
an+1 = f(an)
limits of a sequence
if the limit exists then it converges
if the limit does not exist than it diverges
infinite series
the sum of the terms in a sequence
lim(to inf) (a+b) =
a+b
lim(to inf) cA = (c is a real number)
cA
lim(to inf) AB
AB
increasing
if a(n+1)>an 1,2,3,4,5
noindecreasing
if a(n+1) >= an 1,2,2,3,3,4,4,5
decreasing
if a(n+1)
nonincreasing
if a(n+1)<=an 5,5,4,4,3,3,2,2,1,1
monotonic
if it is either nonincreasing or nondecreasing(moves in one direction)
bounded
if there is a number M such that |a|n<=Mfir all relevant values of n
geometric sequence
have the form r^n or ar^n
what must be true for it to be geometric sequence
a!=0 && r!=0 must have a constant R
convergence and divergence of a geometric sequence
if abs(r) < 1 = converges 0 if r=1 = converges 1 if r<=-1 ||r>1 divgs
bounded monotonic sequence
converges
geometric series
in form ar^k a!=0 and r is != 0
convergence and divergence of a geometric series
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
telescoping series
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
divergence test
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
harmonic series
1/k diverges even though the terms approach 0
integral test
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
p series1/k^p
converges for p>1
diverges for p<=1
ratio test
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
root test
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
comparison test
given ak choose a bk that makes things simple
both must be cont and pos
if bk convgs and ak bk then ak diverges
limit comparison test
given ak choose a bk that makes things simple
both must be cont and pos
lim(to inf) (ak/bk) = L
if 0
steps for determining a test
- try divergence test
- check to see if it is special ie geometric or p
- see if you can integrate it( if so try integral test0
- 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
- if k is a rational function of k use either the comparison test or the limit comparison test
alternating series tests
ak is severything but the (-1)^k
ak>0
ak>ak+1 = ak converges
lim ak = 0
absolute convergence and divergence
both the absolute value and the actual series converge or diverge
conditional convergence and divergance
either of the abs or the regular either diverge or converge
find an n such that the remainder is rem
find ak(like in alt series)
set it as ak+1
estimate alt series
find ak+1
plug that in with the given error
ak+1
