MATH 232 test 2

Sequence {a sub n} converges when

The limit of {a sub n} as n approaches infinity is a finite value

Sequence {a sub n} diverges when

The limit of {a sub n} as n approaches infinity is positive/negative infinity or oscillates

Growth rate most important orders

b^n « n! « n^n

Faster / slower = +- infinity, will not converge

Integral of sin(k*x)dx

-1 / k * cos(k*x)

Integral of cos(k*x)dx

1 / k * sin(k*x)

Integral of sec²(k*x)dx

1/k tan(k*x)

Integral of 1/(1+x²)dx

arctan(x)

Integral of 1/sqrt(1-x²)dx

arcsin(x)

Derivative of cos(x)

-sin(x)

Derivative of sin(x)

cos(x)

Derivative of tan(x)

sec²(x)

Derivative of cot(x)

-csc²(x)

Derivative of sec(x)

sec(x)tan(x)

Derivative of arcsin(x)

1/sqrt(1-x²)

Derivative of arctan(x)

1/(1+x²)

Lim 1/0 →

-+ infinity

Lim 1/+- infinity →

0

Lim e^infinity →

infinity

Lim e^-infinity →

0

Lim ln(infinity) →

infinity

Lim ln(0) →

-infinity

Lim arctan(infinity) →

pi/2

Lim arctan(-infinity) →

-pi/2

Lim (1+(a/n))^(bn) →

e^(a*b)

If |r| < 1, r^infinity →

0

If |r| > 1, r^infinity →

infinity

Lim (ln(n))^(1/n) =

1

Lim n^(1/n) =

1

Lim n!^(1/n) =

infinity

If C > 0, lim C^(1/n) =

1

If p > 0, lim n^(p^(1/n)) =

1

Taylor series for f centered at x = a

f(x) = sum of (f^(n) of a)/n! * (x-a)^n from n=0 to infinity

nth Taylor coefficient for f centered at x = a

c sub n = (f^(n) of a)/n!

Maclaurin series for f centered at x = 0

f(x) = sum of (f^(n) of 0)/n! * x^n from n=0 to infinity

nth Maclaurin coefficient for f centered at a = 0

c sub n = (f^(n) of 0)/n!

Maclaurin series 1/(1-x) =

1 + x + x² + x³ + … = sum of x^n from n=1 to infinity with CI (-1, 1)

Maclaurin series e^x =

1 + x + x²/2! + x³/3! + … = sum of x^n/n! from n=0 to infinity with CI (-infinity, infinity)