|Description: Define a general-purpose
operation that builds a recursive sequence
(i.e. a function on the natural numbers or some other upper
integer set) whose value at an index is a function of its previous value
and the value of an input sequence at that index. This definition is
complicated, but fortunately it is not intended to be used directly.
Instead, the only purpose of this definition is to provide us with an
object that has the properties expressed by seq1 11075
and seqp1 11077.
Typically, those are the main theorems that would be used in practice.
The first operand in the parentheses is the operation that is applied to
the previous value and the value of the input sequence (second
operand). The operand to the left of the parenthesis is the integer to
start from. For example, for the operation , an input sequence
with values 1,
1/2, 1/4, 1/8,... would be transformed into the
output sequence with values 1, 3/2, 7/4,
3/2, etc. In other words, transforms a sequence
means "the sum
of F(n) from n = M to infinity is 2." Since limits are unique
(climuni 12042), by climdm 12044 the "sum of F(n) from n = 1 to
be expressed as
converges) and evaluates to 2 in this example.
Internally, the function generates as its values a set of
ordered pairs starting at , with the first
member of each pair incremented by one in each successive value. So,
the range of is exactly the sequence we want, and we just
extract the range (restricted to omega) and throw away the domain.
This definition has its roots in a series of theorems from om2uz0i 11026
through om2uzf1oi 11032, originally proved by Raph Levien for use
df-exp 11121 and later generalized for arbitrary
Definition df-sum 12175 extracts the summation values from partial
and complete (infinite) series. (Contributed by NM, 18-Apr-2005.)
(Revised by Mario Carneiro, 4-Sep-2013.)