sequences and series

Question 1

what is a sequence

Answer 1

an ordered set of terms

and a rule that specifies them

Question 2

what does the letter U stand for in a sequence

Answer 2

a specific number in the sequence so u1 is the first number ect.

Question 3

how to give a general term for the sequence 1, 3, 5, 7…

Answer 3

ur = 1 + 2(r-1)

with r = 1, 2, 3,

Question 4

how to get a rule from using earlier terms 1, 3, 5, 7…

Answer 4

ur+1 = ur + 2

u1

Question 5

what are rydberg units

Answer 5

H atom energy levels -1/n^2

Question 6

general expression for Fibonacci sequence

Answer 6

nth fibonacci number = (n-1) + (n-2)th

Question 7

what is the fibonacci quarterly

Answer 7

F(n+1)/F(n)
so moving up fibonacci sequence but the numerator is one further up
gets closer to golden ratio

Question 8

what is a series

Answer 8

the sum of the n-term series

Question 9

what is arithmetic progression

Answer 9

sequence with fixed spacing between the terms

Question 10

equation showing the terms of an arithmetic progression

Answer 10

a, a+d, a+2d, a+3d

Question 11

equation for arithmetic sequence using past terms

Answer 11

ur+1 = ur + d
u1 = a

Question 12

general equation for arithmetic equation

Answer 12

ur = a + (r-1)d
r = 1, 2, 3

Question 13

what is d in arithmetic progression

Answer 13

common difference

Question 14

what is geometric progression

Answer 14

sequence with terms a, ax, ax^2, ax3

times by the same thing every time

Question 15

general equation for geometric progression

Answer 15

ur = ax^r-1
r = 1, 2, 3

Question 16

equation for geometric progression using past terms

Answer 16

ur+1 = xur
u1 = a

Question 17

equation for sum of a geometric progression

Answer 17

Sn = a(1-x^n)/(1-x)

Question 18

proof of equation for sum of a geometric progression

Answer 18

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

Question 19

what is x in geometric progression

Answer 19

common ratio

Question 20

what is a in geometric progression

Answer 20

the first term

Question 21

what does capital greek sigma mean

Answer 21

summation running index

Question 22

what is the subscript and superscript above and below the greek sigma

Answer 22

subscript start value

superscript final value

Question 23

equation for the sum of an arithmetic progression

Answer 23

na + n(n-1)d/2

Question 24

what is the limit of an infinite series

Answer 24

the sum of the infinite series
where it trend to
only if it convreges

Question 25

what is the use of converging series in chemistry

Answer 25

to approximate behaviour of a property of a molecule or system
calculate functions from limited data

Question 26

what is a power series

Answer 26

a practically infinite polynomial of x

Question 27

what is R

Answer 27

the radius of convergence

Question 28

where does the power series converge (if at all)

Answer 28

modulus x < R

sometimes R is infinite and the entire series can converge but sometimes it doesn’t

Question 29

example of a power series in chemistry

Answer 29

ideal gas law

approximations to molar specific heat capacity

Question 30

what equation do real gases obey

Answer 30

the virial equation
1 + Bp + Cp^2 + Dp^3
where B C D are constants for a given gas at fixed T

Question 31

equation for molar specific heat capacity approximation

Answer 31

Cp,m = alpha + beta T + theta T

Cpm is a constant independent of temperature

Question 32

what is a maclaurin series

Answer 32

expanding the function f(x) as a power series

approximate expression for the function around the original

Question 33

how to solve a maclaurin series

Answer 33

find the coefficient using differentiation repeatedly
using shorthand index notation for the derivatives
and factorial notation where n! is n factorial

Question 34

general formula for coefficients maclaurin series

Answer 34

f(n) (0) = n!an

an = f(n) (0) / n!

Question 35

general formula for maclauren series

Answer 35

f(x) = f(0) + f’(0)x + 1/2! f’‘(0)x^2 + 1/3! f’’‘(0)x^3

Question 36

info needed to approximate the function at a value of x within the radius of convergence

Answer 36

the values of the function

its derivatives at the origin x = 0

Question 37

what is a taylor series

Answer 37

expanding a function around a point x0

find coefficients from the values of the function and derivatives at x0

Question 38

equation for the taylor series

Answer 38

f(x0) + f’(x0)(x-x0) + 1/2! f’‘(x0)(x-x0)^2 + 1/3!f’’‘(x0)(x-x0)^3

Question 39

general equation for binomial expansion

Answer 39

1 + nx + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3

Question 40

what is the binomial coefficient

Answer 40

n
r
n choose r