series and sequences Flashcards
A sequence can be defined by a recurrence relation too
Un = the nth term of the sequence
U(n+1) = the next term in the sequence
Arithmetic progressions
Adding a fixed amount each time
The first term = a
The difference = d
Nth term formula = a + (n-1)d
Summation = n/2 (2a + (n-1)d)
Geometric progressions
Multiple by a constant each time
the first term = a
the constant = r
Nth term formula = ar^n-1
Summation = a (1 - r^n )/1-r^n
Sequences can be increasing, decreasing and periodic
increasing = each term is larger then the previous
Decreasing = each term in smaller then the previous
Periodic = the terms repeat in a cycle
A convergent series has a sum to infinity
if |r| < 1and n is really big the r^n will be really small
or to put it technically r^n→0
This means (1-r^n) is really close to 1
Sum to infinity of a geometric sequence
a/1-r
A divergent series doesn’t have a a sum to infinity
it continues forever so is impossible to figure out
Pascals Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
binomial formula
e.g (a+bx)^5
(1)(a^5) + (5)(a^4)(b) + (10)(a^3)(b^2)
+ (10)(a^2)(b^3) + (5)(a^1)(b^4) + (1)(b^5)
if n is a fraction or a negative for a binomial expansion
= an infinity sum
has to be in the form (1 + x)^n
1 + nx + (n(n-1)/1x2) (x^2)
some binomial expansions are only valid for certain x values
usually you have state what x is valid for
if n is positive, the binomial expansion is valid for all x values
if n is a negative or a fraction for the expansion (p+qx)^n is valid when |qx/p| < 1
or |x| < |p/q|
partial fractions
e.g
f(x) = x-1 / (3+ x)(1-5x) in the form
A/(3+x) + B/(1-5x)
just times the entire thing by the denominator to get
x-1 = A(1-5x) + B(3+x)
using sigma to find the summation
15
𝛴 (2n+3) 15 is the last term
n+1
