Calculus 3 final

if /an+1/ < /an/ and {an}-->0, the E[(-1)^n * an] converges

alternating series test

the number N is called a lower ____ for a sequence (an) if N < an for all n

bound

a sequence (an) is ____ above if there is a real number M such that an < M for all n

bounded

perpendicular to the axis of the cone, but not including the vertex

circle

a sequence is said to ___ if it has a limit

converge

the ____ of <a,b,c> and <r,s,t> is <bt - cs, cr - at, as - br>

cross product

containing the axis of the cone

degenerate

the ____ of <a,b,c> and <r,s,t> is ar + bs + ct

dot product

9x^2 + y^2 = 9

ellipse

cutting across one nappe of the cone, not perpendicular to the axis

ellipse

x^2/4 + y^2/9 + z^2/16 = 1

ellipsoid

x^2/4 - y^2/9 + z^2/16 = 0

elliptic cone

z= 4x^2 + y^2

elliptic paraboloid

true/false: a geometric series the first term a nd ration r will converge to a/ (1-r), if /r/ > 1

false

true/false: if 0 < an < bn and Ean converges, then Ebn diverges

false

true/false: if P3 is the Maclaurin polynomial of degree 3 for a function f, then P3(0) = f(0), P3(^1)= f(1)(0), P3(^2)(0)= f(^2)(0), and P3(^3)(0)= f(^2)(0)

false

true/false: if an converges to 0, then Ean must necessarily converge

false

true/false: if the ratio test yields the limit 0, then the radius of convergence is 0

false

true/false: if the ratio test yields the limit 2, then the radius of converge is infinity

false

true/false: it is possible to find the cross product of two vectors in a two-dimensional coordinate system

false

true/false: { an + bn} converges to (2L)

false

true/false: {an / bn} converges to (K/L), if bn doesnt equal 0 and K doesnt equal 0

false

true/false:if both Ean and E(-an) converge, then E/an/ must surely converge

false

(π/4)^n

geometric series

cutting into both nappes of the cone, but not including the vertex

hyperbola

y^2 - x^2= 1

hyperbola

z= x^2 - y^2/4

hyperbolic paraboloid

x^2/4 - y^2/9 + z^2/16 = 1

hyperboloid of one sheet

x^2/4 - y^2/9 - z^2/16 = 1

hyperboloid of two sheets

the ____ of a sequence (an) is L if for each E>0, there exits M>0 such that /an - L/ < E whenever n>M

limit

3^n / (4^n + 5)

limit comparison

a sequence is ____ if either (a) each term is greater than or equal to the preceding term, or (b) each term is less than or equal to the preceding term

monotonic

2n / (n + 1)

nth series test

Ean is divergent unless (an)-->0

nth term test

the vector u and v are ____ if u•v=0

orthogonal

1/ n√ n

p-series

if p>1, the E(1/n^p) converges

p-series

(y + 1)^2 = -4(x - 2)

parabola

parallel to one element of the cone

parabola

y^2 + z= 4

parabolic cylinder

Ean converges absolutely if /an+1/ an/ --> L, with L<1

ratio test

if f has n derivatives at c, the nth _____ polynomial for f at c is given by Pn(x) = f(c) + f(^1)(c)(x-c)x^1 + f(^2)(c)(x-c)^2/2 +...+ f(^n)(c)(x-c)^n/n!

taylor

(1/n) - (1/(n + 1))

telescoping series

true/false: a geometric series with first term a and ration r will converge to a/(1-r), if /r/<1

true

true/false: a p-series with p=0.5 will converge

true

true/false: a power series either converges at a single point or at every point in an interval

true

true/false: if 0< an+10 < bn and Ebn converges, then Ean converges

true

true/false: if Ean converges, then an converges to zero

true

true/false: if Ean does not converge, then E/an/ does not converge

true

true/false: if a power series converges at an endpoint of its interval of convergence, its anti-derivative must also converge at that endpoint

true

true/false: if a sequence is bounded and monotonic, then it converges

true

true/false: if the area under f(x) from 1 to infinity is finite, then Ef(n) converges

true

true/false: if the ratio test yields the limit 5/x/, then the radius of convergence is 1/5

true

true/false: if the sequence of partial sums of an converges, then Ean converges

true

true/false: if u and v are vectors in space that are non-zero and non-parallel, then u x v= -v x u

true

true/false: if u doesnt equal 0 and u x v doesnt equal 0 and u x v= u x w, then w=v

true

true/false: if u doesnt equal 0 and u x v= u x w and u • v= u • w, then v=w

true

true/false: if {an} and {bn} both converge to L, and if there exists an N such that for all n>N, an<cn<bn, then {cn} converges to L

true

true/false: {an * bn} converges to (L*K)

true

true/false: {c*bn} converges to (c*K) for any real number c

true

true/false:A taylor polynomial of degree 5 centered at x=7 for a function f has the same slope at x=7 as f does

true

the interval of convergence is ____ for a series that converges between 3 and 9 which converges at 9 but does not converge at 3

(3,9]

A power series centered on 6 converges between ___ and __ if its radius of convergence is 4

2 10

a power series must have radius of convergence ____ and center at ___, if it converges between 5 and 11

3 8

if f has n derivatives at 0, the nth _____ polynomial for f is given by Pn(x)= f(0) + f(^1)(0)x^1 + f(^2)(0)x^2/2 +...+f(^n)(0)x^n/n!

Maclaurin