ap calc bc unit 10 test
alternating series error bound
error=the next term
lim n->infinity
of the sum of the series approaches
nongeometric radius
use ratio test and test endpoints
geometric error bound
|f(x)-P5(x)|= the rest of the series = error
Lagrange Error
|f(x)-pn(x)| < or equal to (|f^(n+1)(c)|/(n+1)!)|x-a|^(n+1)
geometric radius
|x-a|=R r=(x-a)
alternating harmonic
(-1)^n/n converges
1/(1-x)
1+x+x^2+x^3+...+x^n
e^x
1+x+x^2/2!+x^3/3!+...+x^n/n!
nth term test
Diverges if the limit does not equal zero
sin(x)
x-x^3/3!+x^5/5!-x^7/7!+...+(-1)^n(x^2n+1/(2n+1)!)
tan^-1(x)
x-x^3/3+x^5/5-x^7/7+...+x^(2n+1)/(2n+1)!
cos(x)
1-x^2/2!+x^4/4!-...+(-1)^n*x^2n/(2n)!
harmonic
1/n diverges
If the series is geometric and |r|<1,
converges to first term/ 1-r
geometric series test
converges when |r|<1
Maclaurin Series
f(0)+f^1(0)x+f^2(0)x^2/2!+...+f^n(0)x^n/n!
p-series
for 1/n^p, function only converges when p>1
direct comparison test
if bn is larger and its integral converges, an's integral converges if an is smaller and its integral diverages, bn's integral diverges
integral test
if integral converges, series converges
limit comparison test
lim as n approaches infinity... (An) / (Bn) = c, 0<c<infinity, then both the sums of an and bn converge or diverge = 0, then both converge =infinity, then both diverge
alternating series test
lim as n approaches zero of general term = 0 and terms decrease, series converges
RATIO test
lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges if it equals 1, the test is inconclusive
to construct series from known series
substitute, derive, antiderive, multiply series by another series, or multiply series by a function
taylor series
t(a)=f(a)+f^1(a)(x-a)+f^2(a)(x-a)^2/2!+...+f^n(x-a)^n/n!
If the terms don't approach zero,
the series diverges
ln(1+x)
x-(x^2/2)+(x^3/3)...+(-1)^n-1•(x^(n)/n)
