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
