UNIT 10] series convegence/divergence #2 Flashcards
when do you use an alternating series?
when terms swap back and forth between positive and negative values
[they oscillate over and over again over the x-axis]
what is the first condition that needs to be met for the alternating series test to work?
lim n → ∞ (a_n) = 0
[if the limit approaches 0, then the oscillating gets smaller as it approaches infinity, converging]
what is the second condition that needs to be met for the alternating series test to work?
(a_(n+1)) <= (a_n)
[if the next term of n is always smaller, the oscillation will generally become smaller as well]
what does the non-alternating portion of the series in the alternating series test have to have a limit of?
0
[because if the non-alternating portion is 0, then they definitely have no influence on the alternating portion]
when does the ratio test work best?
on complex series
when does the ratio test series converge?
when lim n → ∞ ((a_(n+1))/(a_n)) < 1
when does the ratio test series diverge?
when lim n → ∞ ((a_(n+1))/(a_n)) > 1 or approaches infinity
when is the ratio test series inconclusive
when lim n → ∞ ((a_(n+1))/(a_n)) = 1
how do you find a_(n+1) for the ratio test?
change all “n” in the original expression to “n+1”
which test uses this form to find convergence/divergence: lim n → ∞ |((a_(n+1))/(a_n))|
ratio test
what happens when |a_n| converges?
then ∑ (a_n) is absolutely convergent
what happens when a_n converges but |a_n| diverges?
then ∑ (a_n) is conditionally convergent
[the positive values will get closer to ∞ while the negative values get closer to -∞, making it oscillate forever]
what are the only series that can conditionally converge?
alternating series
[they are the only ones that oscillate]
if a series converges conditionally, then how can the series converge to any real number?
by rearranging the terms of the series
