Week 5 Series

Question 1

what is a series

Answer 1

list of numbers or terms that follow a pattern

Question 2

define an arithmetic series

Answer 2

a series where there is a constant difference between terms

Question 3

what is the general term equation for an arithmetic series

Answer 3

un = u1 + (n-1)d

Question 4

what is the sum equation for an arithmetic series

Answer 4

Sn = n/2 (u1 + un)

Question 5

define a geometric series

Answer 5

series where each term is a constant multiplier of its predecessor

Question 6

what is the general equation for a geometric series

Answer 6

un = ar^n

Question 7

what is the sum equation for a geometric series

Answer 7

Sn = u0(1-r^(n+1)) / 1 - r

Question 8

what happens to geometric series as the number of terms tend to infinity

Answer 8

they either converge or diverge

Question 9

how do you determine whether a geometric series will converge or diverge

Answer 9

you find the modulus of r. If it is less than 1 it converges otherwise it diverges

Question 10

what is the special case and its infinite sum equation for a geometric series

Answer 10

if u0 = 1 then it is a special case where the infinite sum equation is
S = 1 / 1-r

Question 11

define the binomial series

Answer 11

name of the expansion of (a+b)^n

Question 12

what happens to a binomial series when n is negative

Answer 12

the series is infinite and only converges of IxI < 1

Question 13

when is the binomial expansion most useful and why

Answer 13

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

Question 14

what does the taylor and maclaurian series state

Answer 14

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

Question 15

what is the Taylor series equation

Answer 15

f(x) ≈ f(a) + f’(a)/1! (x-a) + … + f^N(a)/N! (x-a)^N

Question 16

what is Maclaurian series equation

Answer 16

f(x) = Σ ( x^n / n! ) f^n(0)

Question 17

why is the Maclaurian series special

Answer 17

it gives rise to the power series

Question 18

what is L’Hopital’s rule used for

Answer 18

used for resolving apparently indeterminable limits

Question 19

what does L’Hopital’s rule state

Answer 19

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

Question 20

what is the general rule formula for L’Hopital’s rule

Answer 20

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)