sequences and series (y1)

sequence

a list of items e.x. 1, 2, 3 ...

series

the sum of a list of terms e.x. 1 + 2 + 3...

sequence and series definitions

increasing and decreasing- each term is greater/lesser than the previous

periodic- the terms repeat in a cycle (u^(n + k) = u^n with an order of k (k unique terms that repeat))

arithmetic sequences

sequences with a common difference between each term

e.x. a, a + d, a + 2d

the nth term is given by u^n = a + (n - 1)d

arithmetic series

sum of the terms of an arithmetic sequence

given by s^n = (n/2)(2a + (n - 1)d) or s^n = (n/2)(a + l)

where n is the number of terms, a is the first term, d is the common difference and l is the last term

proving arithmetic serieses

S^n = a + (a + d) + (a + 2d) ... (a + (n - 2)d) + (a + (n - 1)d)

S^n = (a + (n - 1)d) + (a + (n - 2)d) ... (a + 2d) + (a + d) + a

2S^n = n(2a + (n - 1)d)

therefore S^n = (n/2)(2a + (n - 1)d)

geometric sequences

sequences where each term is r times the last

a, ar, ar^2, ar^3 ...

r is the common ratio

geometric sequence equation

the nth term is given by u^n = ar^(n - 1)

also, u^(k + 1)/u^k = u^(k + 2)/u^(k + 1) = r (pick any two pairs of adjacent terms, and the bigger divided by the smaller is the same for each)

geometric series

the sum of the terms of a geometric sequence

s^n = a(1 - r^n)/(1 - r)

also, by multiplying each side by -1:

s^n = a(r^n - 1)/(r - 1)

(invalid when r = 1 as division by zero)

proving geometric series equation

S^n = a + ar ... ar^(n - 1)

rS^n = ar + ar^2 ... ar^n

S^n - rS^n = a - ar^n

S^n(1 - r) = a(1 - r^n)

therefore S^n = a(1 - r^n)/(1 - r)

convergent and divergent sequences

sequences where the terms tend to a fixed limit or to infinity

convergent and divergent series

series where the sum tends to a fixed limit or to infinity

sum to infinity

the sum of the first n terms for geometric sequences as n approaches infinity

S^inf = a/(1 - r) (only works for convergent sequences, divergent sequences give infinity)

recurrence relations

a way of defining a sequence where each term is given as a function of the previous

e.x. u^(n + 1) = 2u^(n) + 4

sigma notation

∑- denotes the sum of a series

the number below denotes the first value of r and the number above denotes the last, and the equation to the right denotes the equation applied to r