Week 5 Series Flashcards
what is a series
list of numbers or terms that follow a pattern
define an arithmetic series
a series where there is a constant difference between terms
what is the general term equation for an arithmetic series
un = u1 + (n-1)d
what is the sum equation for an arithmetic series
Sn = n/2 (u1 + un)
define a geometric series
series where each term is a constant multiplier of its predecessor
what is the general equation for a geometric series
un = ar^n
what is the sum equation for a geometric series
Sn = u0(1-r^(n+1)) / 1 - r
what happens to geometric series as the number of terms tend to infinity
they either converge or diverge
how do you determine whether a geometric series will converge or diverge
you find the modulus of r. If it is less than 1 it converges otherwise it diverges
what is the special case and its infinite sum equation for a geometric series
if u0 = 1 then it is a special case where the infinite sum equation is
S = 1 / 1-r
define the binomial series
name of the expansion of (a+b)^n
what happens to a binomial series when n is negative
the series is infinite and only converges of IxI < 1
when is the binomial expansion most useful and why
most useful when dealing with (1+x)^n where IxI < 1 because the following approximation can be made for the first few terms
(1+x)^n ≈ 1 + nx
what does the taylor and maclaurian series state
any function that is single valued, continuous and N time differentiable can be well approximated by a polynomial of N+1 derived from the first N differentials at any point in the function’s domain x = a
what is the Taylor series equation
f(x) ≈ f(a) + f’(a)/1! (x-a) + … + f^N(a)/N! (x-a)^N
what is Maclaurian series equation
f(x) = Σ ( x^n / n! ) f^n(0)
why is the Maclaurian series special
it gives rise to the power series
what is L’Hopital’s rule used for
used for resolving apparently indeterminable limits
what does L’Hopital’s rule state
for two functions f(x) and g(x) that are differentiable that both have an infinite limit or zero as a limit then the limit of the quotient of the functions is equal to the limit of the quotient of their derivatives provided the limit exists
what is the general rule formula for L’Hopital’s rule
if limx->a f(x) = limx->a g(x) = 0
then
limx->a f(x)/g(x) = f^(N+1)(a)/g^(N+1)(a)
