sequences and series Flashcards
what is a sequence
an ordered set of terms
and a rule that specifies them
what does the letter U stand for in a sequence
a specific number in the sequence so u1 is the first number ect.
how to give a general term for the sequence 1, 3, 5, 7…
ur = 1 + 2(r-1)
with r = 1, 2, 3,
how to get a rule from using earlier terms 1, 3, 5, 7…
ur+1 = ur + 2
u1
what are rydberg units
H atom energy levels -1/n^2
general expression for Fibonacci sequence
nth fibonacci number = (n-1) + (n-2)th
what is the fibonacci quarterly
F(n+1)/F(n)
so moving up fibonacci sequence but the numerator is one further up
gets closer to golden ratio
what is a series
the sum of the n-term series
what is arithmetic progression
sequence with fixed spacing between the terms
equation showing the terms of an arithmetic progression
a, a+d, a+2d, a+3d
equation for arithmetic sequence using past terms
ur+1 = ur + d u1 = a
general equation for arithmetic equation
ur = a + (r-1)d r = 1, 2, 3
what is d in arithmetic progression
common difference
what is geometric progression
sequence with terms a, ax, ax^2, ax3
times by the same thing every time
general equation for geometric progression
ur = ax^r-1 r = 1, 2, 3
equation for geometric progression using past terms
ur+1 = xur u1 = a
equation for sum of a geometric progression
Sn = a(1-x^n)/(1-x)
proof of equation for sum of a geometric progression
first times everything by x so there is no a term on its own at the beginning and the last term is now ax^n
minus a regular sequence so you end up with -a +ax^n on the right
and xSn - Sn on the left
factorise into brackets and divide so just Sn on the left
what is x in geometric progression
common ratio
what is a in geometric progression
the first term
what does capital greek sigma mean
summation running index
what is the subscript and superscript above and below the greek sigma
subscript start value
superscript final value
equation for the sum of an arithmetic progression
na + n(n-1)d/2
what is the limit of an infinite series
the sum of the infinite series
where it trend to
only if it convreges
what is the use of converging series in chemistry
to approximate behaviour of a property of a molecule or system
calculate functions from limited data
what is a power series
a practically infinite polynomial of x
what is R
the radius of convergence
where does the power series converge (if at all)
modulus x < R
sometimes R is infinite and the entire series can converge but sometimes it doesn’t
example of a power series in chemistry
ideal gas law
approximations to molar specific heat capacity
what equation do real gases obey
the virial equation
1 + Bp + Cp^2 + Dp^3
where B C D are constants for a given gas at fixed T
equation for molar specific heat capacity approximation
Cp,m = alpha + beta T + theta T
Cpm is a constant independent of temperature
what is a maclaurin series
expanding the function f(x) as a power series
approximate expression for the function around the original
how to solve a maclaurin series
find the coefficient using differentiation repeatedly
using shorthand index notation for the derivatives
and factorial notation where n! is n factorial
general formula for coefficients maclaurin series
f(n) (0) = n!an
an = f(n) (0) / n!
general formula for maclauren series
f(x) = f(0) + f’(0)x + 1/2! f’‘(0)x^2 + 1/3! f’’‘(0)x^3
info needed to approximate the function at a value of x within the radius of convergence
the values of the function
its derivatives at the origin x = 0
what is a taylor series
expanding a function around a point x0
find coefficients from the values of the function and derivatives at x0
equation for the taylor series
f(x0) + f’(x0)(x-x0) + 1/2! f’‘(x0)(x-x0)^2 + 1/3!f’’‘(x0)(x-x0)^3
general equation for binomial expansion
1 + nx + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3
what is the binomial coefficient
n
r
n choose r
