MST124 BOOK D UNIT 10

- the term (a_n)¹⁷_n=1 denoted the finite sequence a₁, a₂, a₃, ..., a₁₇
- the term (a_n)^infintiy_n=1 denoted an infitie sequence, with first term a₁ (or just (a_n)) 

Closed form for a sequence
A closed form for a sequence is a formula that defines the general term a_n as an expression involving the subscript n. To specify a sequence using a closed form, two pieces of information are needed:
- the closed form
- the range of values for subscript n 

Reccurence system for a sequence
A recurrence relation for a sequence is an equation that defines each term other than the first term as an expression involving the previous term. To specify a sequence using a recurrence system, three pieces of information are needed 
- the value of the first term
- the recurrence relationship
- the range of values for the subscript n 

Three ways to specify a sequence
A sequence (a_n) can be specified by either giving the values of the first few terms, a closed-form and a subscript range, or a recurrence system (which consists of the value of the first term, a recurrence relation, and a range of values for n) 

- the first term of a sequence a_n is taken to be a₁ unless stated otherwise 

- any sequence formed by adding a fixed number to obtain the next term is called an arithmetic sequence or arithmetic progression
x₁ = a, x_n = x_n-1 + d (n = 2, 3, 4, ...)
a is the first term and d is the common difference 

- for a finite arithmetic sequence, the recurrence system has the form
x₁ = a, x_n = x_n-1 + d (n = 2, 3, 4, ..., N)
where N is the final number in the range of values of n

number of terms = ((last term - first term)/common difference) + 1

Closed form for an arithmetic sequence
The arithmetic sequence with recurrence system
x₁ = a, x_n = x_n-1 + d (n=2,3,4,...)
has the closed form and subscript range
x_n = a + (n-1)d (n=1, 2, 3, ...) 
For a finite arithmetic sequence, don't forget to include the last subscript value N
If you choose to have n start from 0, then the closed form is 
y_n = a +nd (n = 0, 1, 2,...) 

- any sequence obtained by multiplication by a fixed number to obtain the next term is called a geometric sequence or geometric progression 
x₁ = a, x_n = rx_n-1 (n = 2, 3, 4,...)
where a is the first term and r is the number by which you multiply each term to obtain the next (also known as the common ratio)
r is calculated by x_n/x_n-1 

To find the number of terms in a finite geometric sequence, you need to know how many times g₁ has been multiplied by r to obtain g_N   
the equation needed is: r^N-1 = (x_N/x₁)
(take the natural log of both sides and move the one)

Closed form for a geometric sequence
The geometric sequence with recurrence system
x₁ = a, x_n = rx_n-1 (n = 2, 3, 4,...)
has the closed form and subscript range
x_n = ar^n-1 (n = 1, 2, 3, ...) 
- for a finite sequence, you must include the final value in the subscript range (N)

- if you choose the subscript n to start from 0, then the closed form is
y_n = arⁿ (n = 0, 1, 2, ...) 

- the graph of every arithmetic sequence consists of points that lie on a straight line
- the graph of geometric sequences is an exponential growth or decay function
(of the form f(x) = ae^kx where a and k are non-zero constants)
(can also be written as ab^x) 

- a sequence (x_n) is increasing if x_n-1 < x_n for each pair of successive terms x_n-1 and x_n
- it is decreasing if x_n-1 > x_n for each pair 

- suppose all the terms of a sequence (x_n) lie within some interval [-A, A], where A is a fixed +ve number
- we say that the sequence is bounded
- if there is no fixed value of A, however large, then we say that the sequence is unbounded
- in this case we all say that the terms of the sequence become arbitrarily large 

- if the terms of a sequence approach 0 more and more closely, in such a way that they eventually lie within any interval [-h,h], no matter how small the +ve number h is taken to be, we say the sequence becomes arbitrarily small (x_n tends to 0 as n tends to infinity)

- suppose that the terms of a sequence approach a particular number L more and more closely, so they eventually lie within any interval [L-h, L+h], no matter how small the +ve number h is taken to be
- we say that x_n tends to L as n tends to infinity 
- the limit of the sequence is L
- the sequence converges or is convergent to the limit L 

- if a sequence has the property that, whatever +ve number A you take, no matter how large, the terms of (x_n) eventually lie in the interval [A, infinity), then we say that x_n tends to infinity as n tends to infinity 
- this is also true if you swapped + signs for - signs 

The closed form of an arithmetic sequence (x_n) is:
x_n = a + (n-1)d (n = 1,2,3,...)
which can be rearranged to
x_n = (a-d) + nd (n=1,2,3,...)
which is the same as 
x_n = b + nd (n = 1, 2, 3, ...)
where b = a - d

Long-term behaviour of arithmetic sequences
Suppose that (x_n) is an arithmetic sequence with common difference d
- if d > 0, x_n tends to infinity as n tends to infinity
- if d < 0, x_n tends to minus infinity as n tends to infinity
- if d = 0, then (x_n) is a constant sequence 

- closed form of a geometric sequence
x_n = ar^n-1 (n = 1, 2, 3,...)
can be rearranged as
x_n = (a/r)*rⁿ (n = 1, 2, 3,...)
which is the same as 
x_n = crⁿ (n = 1, 2, 3,...)
where c = a/r 
Long-term behaviour of the sequence (rⁿ)
Value of r - Behaviour of rⁿ
r > 1 - increasing, rⁿ tends to infinity as n tends to infinity
r = 1 - constant
0 < r < 1 - decreasing, rⁿ tends to 0 and n tends to infinity 
r = 0 - constant
-1 < r < 0 - alternatives in sign, rⁿ tends to 0 and n tends to infinity
r = -1 - alternates between -1 and 1
r < -1 - Alternates in sign, undbounded 

Multiplying each term of a sequence by a constant
Suppose that (x_n) is an infinite sequence and c is a constant
If c doesn't equal 0 and (x_n):
- is constant, alternates in sign, is bounded, is unbounded, tends to 0
then so is/does (cx_n)
If c > 0 and (x_n):
- is increasing, decreasing, tends to infinity or tends to -infinity
then so is/does (cx_n)
If c < 0 and (x_n):
- is increasing, is decreasing, tends to infinity or tends to -infinity
the (cx_n)
- is decreasing, is increasing, tends to -infinity or tends to infinity 

Suppose that a, c and L are constants
If x_n tends to L as n tends to infinity, then x_n + a tends to L + a as n tends to infinity 
If x_n tends to infinity as n tends to infinity, then x_n + a tends to infinity as n tends to infinity
If x_n tends to -infinity as n tends to infinity, then x_n + a tends to -infinity as n tends to infinity
If x_n tends to L as n tends to infinity, then cx_n tends to cL as n tends to infinity x

- the sum of any finite arithmetic series is
1/2*(number of terms)(first term + last term) 
1/2*n(2a+(n - 1)d)
number of terms = (last term - first term)/common difference + 1

Sum of a finite geometric series
The geometric series with first term a, common ratio r (not equal to 1) and n terms has sum:
(a(1 - rⁿ))/(1 - r) 

Sum of the first n natural numbers
1 + 2 + 3 + ... + n = 1/2*n*(n+1)

Sum of the first n square or cube numbers 
1² + 2² + 3² + ... + n² = 1/6*n*(n + 1)(2n + 1)
1³ + 2³ + 3³ + ... + n³ = 1/4*n²(n + 1)² 

- many infinite series dont have sums
- if the total of a series gets larger as each term is added, without getting closer to any particular number, it doesnt have a sum 

consider any infinite series
a₁ + a₂ + a₃ + ...
Let
s₁ = a₁ 
s₂ = a₁ + a₂
s₃ = a₁ + a₂ + a₃ 
etc
the numbers s₁, s₂, s₃,... etc are called the partial sums of the series, and the infinite series they form is called the sequence of partial sums 

- if the sequence of partial sums of an infinite series converges to a limit, say s, then we call s the sum of the infinite series
- if the sequence of partial sums doesn't converge, then the infinite series doesn't have a sum
- the series with a = 1/2 and r = 1/2 converges to 1, so the infinite series has sum 1 

Sum of an infinite geometric series
The infinite geometric series with first term a (not equal to 0) and common ratio r has
sum a/(1 - r), if -1 < r < 1
no sum if r ≤ -1 or r ≥ 1 

- to find a fraction equivalent to a recurring decimal that has one or more non-zero digits before the recurring part, find a fraction equivalent to the recurring part, and then add this to the number formed by the other digits. 
e.g., s = 0.123123123...
the repeating group is 3 digits long, so multiply s by 10³
1000s = 123.123123123... = 123 + s
999s = 123, so s = 123/999 = 41/333

- the only infinite arithmetic series that has a sum is the infinite series with first term 0 and common difference 0
0 + 0 + 0 + 0 +... = 0 

For any sequence (x_n), the sum
x_p + x_{p + 1} + ... + x_q
(the sum of terms from the pth term to the qth term) is denoted by
sigma^q_{n = p}(x_n)
(n = p to n = q)

Sums of finite arithmetic and geometric series (in sigma notation)
sigmaⁿ_{k = 1}(a + (k - 1)d) = 1/2*n*(2a + (n - 1)d)
sigmaⁿ_{k = 1}(a(1 - rⁿ))/(1 - r), (r does not equal 1) 

Sums of standard finite series (in sigma notation)
sigmaⁿ_{k = 1} (1) = n 
sigmaⁿ_{k = 1}(k) = 1/2*n*(n + 1) 
sigmaⁿ_{k = 1}(k²) = 1/6*n*(n + 1)(2n + 1)
sigmaⁿ_{k = 1}(k³) = 1/4*n²(n + 1)²

Rules for manipulating finite series in sigma notation
sigma^q_{k = p}(cx_k) = c(sigma^q_{k = p}(x_k)) where c is a constant
sigma^q_{k = p}(x_k + y_k) = sigma^q_{k = p}(x_k) + sigma^q_{k = p} (yk)
sigma^q_{k = p} (x_k) = sigma^q_{k = 1} (x_k) - sigma^{p - 1}_{k = 1} (x_k) (where 1 < p ≤ q)

- to write an infintie series in sigma notation, you write the infinity symbol in place of the upper limit 

Sum of an infinite geometric series (in sigma notation)
If -1 < r ≤ 1, then: sigma^infinity_{n = 1}(ar^n-1) = a/(1 - r) 

(a + b)⁰ = 1
(a + b)¹ = a + b
(a + b)² = a² + 2ab + b² 
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
(a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵
(a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶ 
- the powers in each term on the RHS add up to the power of the brackets on the LHS 
- each coefficient on the RHS is the sum of the two adjacent coefficients in the line above (when in a pyramid formation, the two numbers diagonally above left and right)

- the coefficients for any of the formulas can be found in Pascal's triangle
- the array has 1s down each edge, and each of its other numbers is the sum of the two adjacet numbers in the line above
- it can be continues indefinitely 

- numbers in Pascal's triangle are called the binomial coefficients
- if the rows of Pascal's triangle are numbered as row 0, row 1, row 2 and so on, and the coefficients within each row are numbered from left to right as coefficient 0, coefficient 1, coefficient 2 and so on
- coefficient k in row n is denoted by ⁿC_k 
^rowC_coefficient 

The binomial theorem
For any natural number n,
(a + b)ⁿ = ⁿ C₀aⁿ + ⁿC₁a^n-1b + ... + ⁿ C_n-1ab^n-1 + ⁿC_nbⁿ 

- for any natural number n, the product of all the natural numbers up to and including n is called the factorial of n and is denoted by n!
0! = 1 
n! = n(n - 1)! (n = 1, 2, 3,...)
c₀ = 1, c_n = nc_n-1 (n = 1, 2, 3,...)

If n and k are integers with 0 ≤ k ≤ n, then
ⁿC_k = (n!)/(k!(n-k)!)

If n and k are integers with 0 < k ≤ n, then
ⁿC_k = (n(n - 1)...(n - k + 1))/(k(k - 1)...1)
- start with the integer n on the top and the integer k on the bottom, and for each of these integers you keep multiplying by the next integer down until you have k factors in total 

- the binomial coefficients in each row of Pascal's triangle are the same no matter whether you read them from left to right or right to left
ⁿC_k = ⁿC_{n - k} 

The binomial theorem (restated)
For any natural number n, 
(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^{n - 1}b + ... + ⁿC_{n - 1}ab^{n - 1} + ⁿC_nbⁿ 
where
ⁿC_k = (n!)/(k!(n - k)!) (k = 0, 1, 2,...,n)

The binomial theorem (sigma notation)
For any natural number n,
(a + b)ⁿ = sigmaⁿ_k=0(ⁿ C_ka^n-kb^k) = sigmaⁿ_k=0(n!)/(k!(n-k)!)a^n-kb^k 

(1 + x)ⁿ = 1 + ⁿC₁x + ⁿC₂x² + ... + ⁿC_kx^k + ... + xⁿ 
= 1 + nx + ((n(n-1)/2!)x² + ((n(n-1)(n-2)/3!)x³ + ... + xⁿ