UNIT 10] series convergence/divergence Flashcards
what does a geometric series look like?
∑ ∞ n=0 (a)(r^n)
when is a geometric series convergent?
when |r|<1
when is a geometric series divergent?
when |r|>=1
how do you find the sum of a geometric series if it converges?
(a)/(1-r)
what does a series that the nth-term test works on look like?
∑ ∞ n=0 (a_n)
can an nth-term test be used to find convergence?
no
when is a nth-term series divergent?
when lim n→∞ (a_n) =/= 0, or when it does not exist.
what happens in an nth term series if the limit is 0?
there is no conclusion
what does a series that the integral test can be used on look like?
∑ ∞ n=N (a_n)
when is an integral test series convergent?
∫[N, ∞] (f(x))dx = L
when is an integral test series divergent?
∫[N, ∞] (f(x))dx = ∞
what does the graph of f(x) have to behave like for an integral test?
f(x) has to be positive, decreasing, and continuous
what is the behavior of the series like when doing an integral test?
the behavior of the series and integral are the same
what does a p-series look like?
∑ ∞ n=N (1)/(n^p)
when is a p-series convergent?
when p>1
when is a p-series divergent?
when 0<p<=1
what is a p-series called when p=1?
a harmonic series
does the harmonic series always diverge or converge?
diverge
if a series is an airplane, what is divergence?
when the plane lands
if a series is an airplane, what is convergence?
when the plane is gaining altitude
what does a series that you can use the direct comparison test on look like?
∑ (a_n)
when is a direct comparison test series convergent?
when a_n <= c_n
when is a direct comparison test series divergent?
when a_n>= c_n
what are the two functions being compared in a direct comparison test?
∑ (a_n) (original) and ∑ (c_n)
in a direct/limit comparison test, what do the nth term expressions need to be
> 0, positive
in a direct/limit comparison test, what do you need to know about the new series?
if it converges or diverges
when is a limit comparison series convergent
when lim n→∞ ((a_n)/(c_n)) = 0
when is a limit comparison test series divergent?
when lim n→∞ ((a_n)/(c_n)) = ∞
