Test 2 Flashcards
d’Alembert’s ratio test
L=lim(Fk+1)/Fk < 1 Lim is k towards infinity If L<1, series converges If L>1, series diverges If L=1, no help
When to use maclaurin series vs taylor series
Maclaurin series approximates around zero. Taylor series approximates around a point ‘a’
Eulers formula
e^jθ = cos(θ) + jsin(θ)
Exponential form of complex numbers
z=r*e^jθ
Complex trig answers: exponential form and expanding it
cos(z) = 1/2 * (e^jz + e^-jz)
sin(z) = 1/2j * (e^jz - e^-jz)
Sub z for x + jy and use trig and hyperbolic complex rules to solve
Converting between hyperbolic and exponential functions
cosh(θ) = 1/2 * (e^θ + e^-θ)
sinh(θ) = 1/2 * (e^θ - e^-θ)
Converting between complex trig and hyperbolic functions
cosh(jx) = cos(x) cos(jx) = cosh(x) sinh(jx) = jsin(x) sin(jx) = jsinh(x)
Working out ln(z)
ln(z) = ln(re^jθ)
= ln(r) + ln(e^jθ)
= ln(r) + jθ
Working out e^z = a
Take logs of both sides
z = ln(a)
Pretend a is a complex number and rewrite using ln(z)
Complex powers
Rewrite the thing being raised in the exponential form. One of them will times by the complex power while you rewrite the other as e^ln(x) and stick the j in front
Powers of trig functions and complex variable functions
Revise this
Partial fractions: irreducible quadratic on the bottom, repeated factor on the bottom
IQ: Ax + B above the quadratic term
RF: Once with the factor by itself, then again with it squared
Volume of parallelepiped
(axb).c |
Taylor series formula around point ‘a’
f(a) + f’(a)(x-a) + f’‘(a)/2! (x-a)^2 + f’’‘(a)/3! (x-a)^3
Writing a sequence that goes up like 1, 3, 5, etc
Use 2n + 1
