EngMaths 245 - Week 1 to Week 2 - 11.1 to 11.2

Define a sequence and give an example

• A sequence is a list of numbers

• e.g $a_1, a_2, a_3, …, a_n, …$

Describe the two ways you can denote a sequence

• $$(a_n), (a_n)^{\infty}_{n = 1}$$

• e.g $(n + 3)^{\infty}_{n=1}$ would give:

  • 1, 4, 7, 10, ….

  • superscript is the final n

  • subscript is the initial n

• e.g $a_n = a_{n-1} + a_{n-2}$

Describe what is meant by the limit of a sequence

• A sequence with a limit L will get arbitrarily closer to that limit as n increases

Describe the condition for the convergence of a sequence

• A sequence ($a_n$)

• For any $\epsilon$ > 0:

  • if a natural number N exists with the following property:

    • n $\geq$ N

  • then | $a_n$ - L | < $\epsilon$

• otherwise it diverges

Describe the condition for the divergence of a sequence of a sequence

• A sequence ($a_n$)

• For all numbers M:

  • if a natural number N exists with the following property:

    • n $\geq$ N

  • then $a_n$ $\geq$ M

Derive the proof relating the limit of a function to the limit of a sequence

• if $\lim_{x \to \infty} f(x) = L$ and

  • $a_n = f(n)$ then

  • $$\lim_{n \to \infty} a_n = L$$

State the limit laws referring to the limit of two sequences that have the following operations applied to each other:

• summed

• subtracted

• divided

• multiplied

% 1) Sum Law

$$\lim (a_n + b_n) = \lim (a_n) + \lim (b_n)$$

% 2) Difference Law

$$\lim (a_n - b_n) = \lim (a_n) - \lim (b_n)$$

% 4) Product Law

$$\lim (a_n b_n) = \lim (a_n) \lim (b_n)$$

% 5) Division (Quotient) Law

$$\lim (a_n / b_n) = \frac{\lim(a_n)}{\lim(b_n)}, \quad \lim(b_n) \neq 0$$

State the limit law regarding the limit of the product of a sequence and a constant

• $$\lim (c\cdot a_n) = c\cdot \lim(a_n)$$

State the limit law regarding the limit of the power of a sequence and a variable p > 0, a_n > 0

• $$\lim (a_n^p) = (\lim(a_n))^p$$

Describe the squeeze theorem regarding sequences

• if $a_n \leq b_n \leq c_n$ and $lim(a_n) = lim(c_n)</p><ul><li><p>$lim(b_n) = lim(a_n) = lim(c_n)$$

State the limit law regarding $\lim_{n \to \infty} |a_n| = 0$

• if $lim_{n \to \infty} |a_n| = 0$ then $lim_{n \to \infty} a_n = 0$

State the limit law of sequences relating the limit of a sequence and a function of that sequence as an input

Define a monotone sequence

Define a bounded sequence

Relate the monotone and bounded properties of a sequence to it convergence

Describe and define what is meant by the series and partial sums of a sequence

Relate the convergence of a series to its partial series

Define what is meant by a geometric series

• sum of numbers in a sequence where

  • each term is found by multiplying the previous term by a fixed, non-zero number

    • that number is called the common ratio r

State the formula for the series of a geometric sequence

• $$\Sigma^{\infty}_{n = 1} a\cdot r^{n-1}$$

  • a = first term

  • r = common ratio

Derive the equation for the sequence of partial sums of a geometric series

Derive the condition for the convergence of a geometric series

Derive and state the series laws related to different series:

• addition

• subtraction

• constant