Numerical Sequences

Numerical Sequences: General Information

• A real number sequence is a function u: N -> R
• u(n) = un is the image of integer n by u, and also the general term of sequence u
• Numerical sequences can be defined explicitly as un = f(n) or by recurrence as u(n+1) = f(un), with u0 being given

Numerical Sequences: Examples

• Arithmetic sequence: u(n+1) = un + r
• Geometric sequence: u(n+1) = q * un

Graphical Representation

• The graphical representation of a sequence in an orthogonal coordinate system is the set of points with coordinates (n; un)
• Given a sequence defined by recurrence u(n+1) = f(un), the first terms can be constructed on the x-axis

Variations

• A sequence (un) is increasing if, for all n in N, un ≤ u(n+1)
• A sequence (un) is decreasing if, for all n in N, un ≥ u(n+1)
• A sequence (un) is monotone if it's either increasing or decreasing

Studying the variations

• Study the sign of u(n+1) - un
• If un > 0, compare u(n+1) / un to 1
• If un = f(n), study the variations of the function f

Studying the variations: Examples

• un = n^2 gives the result: u(n+1) - un = (n+1)^2 - n^2 = 2n + 1 > 0, which implies that (un) is increasing
• un = 1/(n+1) becomes u(n+1) - un = -1 / ((n+2)(n+1)) < 0; hence, (un) is decreasing

Majorized, Minorized, and Bounded Sequences

• A sequence (un) is majorized if there exists a real number M such that, for all n in N, un ≤ M
• A sequence (un) is minorized if there exists a real number m such that, for all n in N, un ≥ m
• A sequence (un) is bounded if it is both majorized and minorized

Majorized, Minorized, and Bounded Sequences: Examples

• un = sin(n) is bounded because -1 ≤ un ≤ 1
• un = n^2 is minorized by 0, but not majorized

Limits

Finite Limit

• A sequence (un) converges to a real number l if every open interval containing l contains all un values from a certain rank onward, denoted as lim (n→+∞) un = l

Finite Limit: Example

• un = 1/(n+1), this becomes lim (n→+∞) un = 0

Infinite Limit

• A sequence (un) tends towards +∞ if every interval of the form ]A ; +∞[ contains all un values from a certain rank onward, which can be denoted as lim (n→+∞) un = +∞
• A sequence (un) tends towards -∞ if every interval of the form ]-∞ ; A[ contains all un values from a certain rank onward, which can be denoted as lim (n→+∞) un = -∞

Infinite Limit: Examples

• un = n^2, therefore lim (n→+∞) un = +∞
• un = -n, therefore lim (n→+∞) un = -∞

Operations on limits

• If lim(un) = l and lim(vn) = l', then lim(un + vn) = l + l'
• If lim(un) = l and lim(vn) = +∞, then lim(un + vn) = +∞
• If lim(un) = -∞ and lim(vn) = l, then lim(un + vn) = -∞
• If lim(un) = +∞ and lim(vn) = +∞, then lim(un + vn) = +∞
• If lim(un) = -∞ and lim(vn) = -∞, then lim(un + vn) = -∞
• If lim(un) = +∞ and lim(vn) = -∞, then lim(un + vn) has an indeterminate form

Operations on limits; multiplication

• If lim(un) = l and lim(vn) = l', then lim(un × vn) = l × l'
• If lim(un) = l > 0 and lim(vn) = +∞, then lim(un × vn) = +∞
• If lim(un) = l > 0 and lim(vn) = -∞, then lim(un × vn) = -∞
• If lim(un) = l < 0 and lim(vn) = +∞, then lim(un × vn) = -∞
• If lim(un) = l < 0 and lim(vn) = -∞, then lim(un × vn) = +∞
• If lim(un) = 0 and lim(vn) = ±∞, then lim(un × vn) has an indeterminate form
• If lim(un) = ±∞ and lim(vn) = 0, then lim(un × vn) has an indeterminate form
• If lim(un) = +∞ and lim(vn) = +∞, then lim(un × vn) = +∞
• If lim(un) = -∞ and lim(vn) = -∞, then lim(un × vn) = +∞
• If lim(un) = +∞ and lim(vn) = -∞, then lim(un × vn) = -∞

Operations on limits; division

• If lim(un) = l and lim(vn) = l' ≠ 0, then lim(un / vn) = l / l'
• If lim(un) = l and lim(vn) = ±∞, then lim(un / vn) = 0
• If lim(un) = ±∞ and lim(vn) = l' ≠ 0, then lim(un / vn) = ±∞ (same sign as 1/l')
• If lim(un) = l ≠ 0 and lim(vn) = 0+, then lim(un / vn) = ±∞ (same sign as l)
• If lim(un) = l ≠ 0 and lim(vn) = 0-, then lim(un / vn) = ±∞ (same sign as -l)
• If lim(un) = 0 and lim(vn) = 0, then lim(un / vn) has an indeterminate form
• If lim(un) = ±∞ and lim(vn) = ±∞, then lim(un / vn) has an indeterminate form

Comparison Theorems

• If for all n ≥ p, un ≥ vn and lim (n→+∞) vn = +∞, then lim (n→+∞) un = +∞
• If for all n ≥ p, un ≤ vn and lim (n→+∞) vn = -∞, then lim (n→+∞) un = -∞

Gendarmes Theorem

• If for all n ≥ p, vn ≤ un ≤ wn and lim (n→+∞) vn = lim (n→+∞) wn = l, then lim (n→+∞) un = l

Gendarmes Theorem: example

• With the result of un = sin(n)/n, we know that -1/n ≤ sin(n)/n ≤ 1/n
• As lim (n→+∞) -1/n = 0 and lim (n→+∞) 1/n = 0, therefore lim (n→+∞) sin(n)/n = 0

Sequences and Functions

• If lim (n→+∞) un = +∞ and lim (x→+∞) f(x) = l, then lim (n→+∞) f(un) = l
• If lim (n→+∞) un = l and lim (x→l) f(x) = l', then lim (n→+∞) f(un) = l'

Sequences and Functions examples

• If un = 1/n and f(x) = e^x, then lim (n→+∞) un = 0 and lim (x→0) f(x) = 1
• As a result, lim (n→+∞) e^(1/n) = 1

Particular Sequences

Arithmetic Sequences

• An arithmetic sequence is defined by u(n+1) = un + r, where r is the common difference
• Properties: un = u0 + nr and un = up + (n-p)r

Sum of terms

• S = u0 + u1 + ... + un = (n+1) × (u0 + un) / 2
• S = (number of terms) × (first term + last term) / 2

Arithmetic sequences: example

• 1 + 2 + ... + n = n × (1+n) / 2 = n(n+1) / 2

Geometric Sequences

• A geometric sequence is defined by u(n+1) = q * un, where q is the common ratio
• Properties: un = u0 × q^n and un = up × q^(n-p)

Sum of terms

• If q ≠ 1, S = u0 + u1 + ... + un = u0 × (1-q^(n+1)) / (1-q)
• If q = 1, S = (n+1)u0

Geometric Sequences: example

• 1 + 2 + 4 + 8 + 16 = 1 × (1-2^5) / (1-2) = 31

Limits

• If q > 1, then lim (n→+∞) q^n = +∞
• If q = 1, then lim (n→+∞) q^n = 1
• If -1 < q < 1, then lim (n→+∞) q^n = 0
• If q ≤ -1, then (q^n) has no limit
• lim (n→+∞) a^n / n! = 0