unit 9 Flashcards
n-th term test
try this first- only tells if it diverges. if lim is not 0, the series diverges. if lim = 0 test is inconclusive
Geometric series test
series converges if |r|<1. sum is given by a/1-r
integral test
if you can take the integral of the series. if f is continuous, decreasing, and positive then both either converge or diverge
p-series test
if p>1 it converges, if p<=1 it diverges
direct comparison test
compare series to one just bigger. 0<an<=bn if Bn converges, An converges
limit comparison test
good for rational/poly. lim an/bn does not = 0 both an and bn converge or diverge
alternating series test
converges if alternating, decreasing, and limit = 0.
ratio test
lim |an+1/an| < 1 it converges. lim |an+1/an| > 1 diverges.
lim=1 test inconclusive
root test
if lim n root (|an|) <1 converge
lim >1 or DNE, diverge
lim=1, inconclusive
Taylor and Maclaurin polynomials
Maclaurin- centered at zero. form: P(n)=f(c)+f’(c)(x-c)/1!+f’‘(c)(x-c)^2/2!…
actual error
find the polynomial. Do actual number - polynomial number
error bound
if series is alternating and decreasing, error bound can be found by taking the absolute value of the n+1 term.
Lagrange error bound
|Pn(x) - sum An|<= f^n+1(z)(x-a)^n+1/(n+1)!
Power series
Sum AnX^n = a/1-r, |r|<1
or
sum An(x-c)^n = x=c
-check end pointsc
common Maclaurin series
ex=x^n/n!
sin x = (-1)^n x^2n+1 / (2n+1)!
cos x= (-1)^n x^2n / (2n)!
1/1-x = x^n
ln (1+x) = (-1)^n+1 x^n/n
