Calculus II: sequences & series
sequences & series calculus

sequences Properties

1. monotonic sequence

A sequence is monotonic if and only if it is either entirely increasing or decreasing.

Given any sequence {a_n} we have the following:
• We call the sequence increasing if a_n < a_n+1 for every n.

• We call the sequence decreasing if a_n > a_n+1 for every n.

• If {a_n} is an increasing sequence or {a_n} is a decreasing sequence, the sequence is said monotonic.

• If there exists a number m such that m ≤ a_n for every n we say the sequence is bounded below. The number m is sometimes called a lower bound for the sequence.

• If there exists a number M such that a_n ≤ M for every n we say the sequence is bounded above. The number M is sometimes called an upper bound for the sequence.

• If the sequence is both bounded below and bounded above we call the sequence bounded.

2. Limit of sequence

The limit of a sequence is the limit of its general term.

For the sequence {a_n}:

If lim a_n = L    (L)
n → ∞  

then, the sequence { a_n } is convergent.

else

if the limit L does not exist, or infinite, the sequence is divergent.

An alternating sequence {(-1)ⁿbn} with bn > 0 is:

• convergent to 0 , if lim [n → ∞] bn = 0
• divergent if lim [n → ∞] bn≠ 0 .

3. Properties of sequence

• lim [n → ∞] (a_n ± b_n) = lim [n → ∞] a_n ± lim [n → ∞] b_n

• lim [n → ∞] (c a_n) = c lim [n → ∞] a_n

• lim [n → ∞] (a_n x b_n) = lim [n → ∞] a_n x lim [n → ∞] b_n

• lim [n → ∞] (a_n / b_n) = lim [n → ∞] a_n / lim [n → ∞] b_n,
        provided lim [n → ∞] b_n ≠ 0.

• lim [n → ∞] (a_n)^p = (lim [n → ∞] a_n)^p,
        provided a_n ≥ 0.

4. Related theorems of sequence

Theorem 0

If {a_n} is bounded and monotonic then {a_n} is convergent.

Note that if a sequence is not bounded and/or not monotonic does not say that it is divergent.

Theorem 1

Given the sequence {a_n} if we have a function f(x) such that f(n) = a_n and
lim [x → ∞] f(x) = L then lim [n → ∞] a_n = L

Squeeze Theorem

If a_n ≤ c_n ≤ b_n for all n ≥ N

for some N and lim [n → ∞] a_n = lim [n → ∞] b_n = L then
lim [n → ∞] c_n = L .

Theorem 2

If lim [n → ∞] |a_n| = 0 then
lim [n → ∞] a_n = 0.

Theorem 3

The sequence {rⁿ} [ n = 0 → ∞]
does not have a limite when r ≥ 1,
has a limite 0 when - 1 ≤ r ≤ 1
has a limite 1 when r = 1
diverges for r ≥ 1

Theorem 4

For the sequence {aⁿ} if both lim [n → ∞] a_2n = L and
lim [n → ∞] a_2n+1 = L then {aⁿ} is convergent and
lim [n → ∞] a_n = L.