Exam 4 Flashcards
Extended Ratio Test
the limit as n approaches infinity of the absolute value of a of n+1 over the absolute value of a of n. If it is less than one it converges, if it is greater the one it diverges.
Extended Root Test
The limit as n approaches infinity of the nth root of the absolute value of a of n. If it is less than one it converges, if it is greater than one it diverges.
Power Series
a series which has variable terms.
Interval of Convergence
the set of all values of x for which a power series converges.
Radius of Convergence
half the length of the interval of convergence.
The variable x can be:
any real number
Because x can be any real number we must use tests that:
Work for any type of series
-Extended Ratio Test
-Extended Root Test
General form of Power Series Representation
For x is greater than negative one but less than one: 1/(1-x) = the series from n=0 to infinity of x^n = 1 + x + x^2 + x^3 + …
Does differentiating of integrating a power series change its interval of convergence or radius of convergence?
No
Taylor Series
f(x) = f(a) + (f’(a)/1!)(x-a) + (f’‘(a)/2!)(x-a)^2 +…
Maclaurin Series (Taylor series for a=0)
f(x) = f(0) + (f’(0)/1!)x + (f’‘(0)/2!)(x^2) + …
Power Series Expansion : f(x) = e^x
the series from n=0 to infinity of (x^n)/n! for x is greater than negative infinity but less than infinity.
Power Series Expansion: f(x) = sinx
the series from n=0 to infinity of (-1)^n ((x^2n+1)/(2n+1)!) for x is greater than negative infinity but less than infinity.
Power Series Expansion: f(x) = cosx
the series from n=0 to infinity of (-1)^n ((x^2n)/(2n)!) for x is greater than negative infinity but less than infinity.
nth degree Taylor polynomial
Tn(x) = f(a) + (f’(a)/1!)(x-a) + (f’‘(a)/2!)(x-a)^2 + … + (f^(n) (a)/n!)(x-a)^n
Taylor’s formula/Taylor’s Inequality
|Rn(x)| is less than or equal to (M/(n+1)) *(|x-a|^n+1) where |f^(n+1) (x)| is less than or equal to M.
Arc length - function of y
the integral from c to d of the square root of 1 + (dx/dy)^2 dy
Arc length - function of x
the integral from a to b of the square root of 1 + (dy/dx)^2 dx
Area of a surface of revolution - function of x
the integral from a to b of 2pir times the square root of (1+ (dy/dx)^2) dx
Area of a surface of revolution - function of y
the integral from c to d of 2pir times the square root of (1+ (dx/dy)^2) dy
Parameter
variable t
Cycloid
curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line.
First derivative of parametric equations
(dy/dx) = (dy/dt)/(dx/dt)
Second derivative of parametric equations
(d^2 y/dx^2) = (d/dt(dy/dx))/(dx/dt)
Point-slope form
(y-y1) = m(x-x1)
Area under a Curve - Parametric equations
the integral from alpha to theta of g(t) * f’(t) dt
Arc Length - Parametric equations
the integral from alpha to theta of the square root of ((dx/dt)^2 + (dy/dt)^2) dt
Surface Area - Parametric equations
the integral from alpha to theta of 2pir * square root of ((dx/dt)^2 + (dy/dt)^2) dt
