Unit 7/8 Calc BC Flashcards
telescoping series
usually used with basic fractions (n is in numerator), Sn = first term - lim n approaches infinity of last term (determine terms by writing out series and look for patterns)
convergence of a geometric series
sum of a(r)^n, diverges if |r| >= 1, converges if
0 < |r| <1
sum is a / (1 - r)
limit of the nth term of a convergent series
if the series converges, the limit as An goes to infinity is 0 (DOES NOT MEAN if limit equals 0, a series converges)
nth-term test for divergence
if the limit as An does not equal zero, then the series must diverge
the integral test
if f(x) if postive, continuous, and decreasing for x >= 1, then the integral as the series goes from 1 to infinity either both converge or both diverge
convergence of a p-series
converges if p > 1, diverges if 0 < p <= 1
maximum error
if An converges to S, then the remainder Rn is between 0 < Rn < integral from n to infinity of An
Sum = Sn + Rn
direct comparison test
Bn (big series) >= An (small series) > 0
If Bn converges, then An converges
If An diverges, then Bn diverges
(compare which function is bigger, which determines which series is bigger)
limit comparison test
if both An and Bn are positive (An is original function, Bn is ‘simplified function’)
and lim n -> infinity (An/Bn) = L (finite and positive),
then both series either converge or diverge
alternating series
(-1)^n-1(An) or (-1)^n(An) converges if
lim n-> infinity An = 0 AND
the sequence An is decreasing and positve (0 < An+1 < An)
alternating series remainder
if an alternating series converges, than the absolute value of the remainder |Rn| is <= An+1
|Sn - Sn+1| is equal to the error in the sum
definition of absolute and conditional convergence
An absolutely converges if |An| converges
An conditionally converges if An converges BUT |An| diverges
ratio test
(used mostly with n! or x^n)
find lim n -> infinity (| An+1 / An | ) = L
L < 1, then series converges absolutely
L > 1, then series diverges
if L = 1, then must try another test (absolutely converges)
root test
find the limit n -> infinity (nth root of |An| ) = L :
L < 1, then series converges absolutely
L > 1, then series diverges
if L = 1, then must try another test (inconclusive)
absolute value theorem for sequences
if lim n -> infinity |An| = 0, then lim n–> infinity An = 0
(just for sequences, not for series)
