Series Flashcards
(16 cards)
nth term test
lim n –> infinity an ≠ 0
series diverges
lim n –> infinity an = 0
no info / try another test
Geometric series test
∑ a (r)^n
r < 1 –> converges
r > 1 –> diverges
r = 1 –> weird series (nth term test)
Formula for geometric sum
initial term / ( 1 - r )
P-series test
∑ a / (n^p)
p < 1 –> diverges
p > 1 –> converges
p = 1 –> diverge (harmonic)
Alternating series test
1) terms must alternate signs
2) lim (n –> infinity) an = 0
3) terms decrease in absolute value
terms don’t alternate –> don’t use this test
lim (n –> infinity) an ≠ 0 –> diverges (nth term)
doesn’t decrease in |value| –> no info
1 , 2 , and 3 are all true –> converges
Direct comparison test
Compare to g-series or p-series
0 ≤ given ≤ simple conv. –> converges
given > simple conv. –> no info (use L.C.T.)
given < simple div. –> no info (use L.C.T.)
given > simple div. –> diverges
Limit comparison test
lim (n –> infinity) [ given / simple ]
can also be [ simple / given ]
limit is finite/positive –> both series conv. or div.
limit is 0 or DNE –> no info
Integral test
f(x) is continuous, positive, and eventually decreasing
∫ (# to infinity) f(x) dx conv. –> ∑ converges
∫ (# to infinity) f(x) dx div. –> ∑ diverges
Root test
Find:
lim (n –> infinity) |an|^(1/n)
limit < 1 –> converges (absolutely)
limit > 1 –> diverges
limit = 1 –> test fails
Ratio test
lim (n –> infinity) |(an + 1) / (an) |
limit < 1 –> converges
limit > 1 –> diverges
limit = 1 –> no info
(Big 5) Geometric polynomial formula
1 / (1 - x)
(n = 0) ∑ x^n
(Big 5) Exponential polynomial formula
e^x
(n = 0) ∑ 1 / n! * x^n
(Big 5) Logarithmic polynomial formula
ln (x)
(n = 1) ∑ -1^(n+1) / n * (x - 1)^n
(Big 5) Cosine polynomial formula
cos (x)
(n = 0) ∑ (-1)^n / (2n) ! * x^2n
(Big 5) Sine polynomial formula
sin (x)
(n = 0) ∑ (-1)^n / (2n + 1) ! * (x)^(2n+1)
Taylor vs Maclaurin series
Maclaurin is centered at zero while Taylor is centered at #
