Sequences and series definitions

Question 1

modulus function

Answer 1

for x∈ℝ we define |x|∈ℝ by the formula:

|x| = x if x≥0 or -x if x<0

Question 2

sequence tending to infinity

Answer 2

A sequence (aₙ) tend to infinity if given any real number, A>0, there exists a point in the sequence, N∈ℕ, such that (aₙ)>A wherever n>N

Question 3

sequence tending to minus infinity

Answer 3

A sequence (aₙ) tend to minus infinity if given any real number, A>0, there exists a point in the sequence, N∈ℕ, such that (aₙ) is less than -N wherever n>N

Question 4

Sequence converging to a point

Answer 4

A sequence (aₙ) converges t a real number l if given any small number, ε>0, there exists a point in the sequence, N∈ℕ, such that |(aₙ)-l| is less than epsilon wherever n>N

Question 5

Bounded sequence

Answer 5

A sequence (aₙ) is:

1) bounded above if there exists M∈ℝ such that (aₙ)≤M for all n∈ℕ
2) bounded below if there exists M∈ℝ such that (aₙ)≥M for all n∈ℕ
3) bounded if its both bounded above and below

Question 6

subsequence

Answer 6

a subsequence of a sequence (aₙ) takes the form aₙ₁, aₙ₂, aₙ₃… where n₁,n₂,n₃ is a strictly increasing sequence of ℕ

Question 7

Monotone sequence

Answer 7

A sequence (aₙ) is:

1) increasing if (aₙ₊₁)≥(aₙ)
2) strictly increasing if (aₙ₊₁)>(aₙ)
3) decreasing if (aₙ₊₁)≤(aₙ)
4) strictly decreasing if (aₙ₊₁)

Question 8

series

Answer 8

the sum of all the terms in a sequence, it is an expression of the form:
Σ(aₙ) = a₁ + a₂ + a₃ + … + aₙ

Question 9

converging series

Answer 9

A series Σ∞ₙ₌₁(aₙ) converges to a real number s if its sequence of partial sums Sₙ = Σⁿₖ₌₁(aₖ) converges to s

Question 10

geometric series

Answer 10

for r∈ℝ. The series Σ∞ₙ₌₀(rⁿ)= 1 + r + r² + … + rⁿ is called a geometric series.

Question 11

geometric series formula

Answer 11

SN = (1-rᴺ⁺¹)/1-r

Question 12

geometric series to infinity formula

Answer 12

1/1-r

|r|<1

Question 13

absolute convergence

Answer 13

a series Σ∞ₙ₌₁(aₙ) converges absolutely if Σ∞ₙ₌₁|(aₙ)| converges

Question 14

conditional convergence

Answer 14

a series Σ∞ₙ₌₁(aₙ) converges but not absolutely

Question 15

The cauchy product

Answer 15

the product of two infinite series

Question 16

power series

Answer 16

a series of the form Σ∞ₙ₌₀(aₙ)(xⁿ) where (aₙ) is a sequence of real numbers(fixed for individual series) and x is a variable.

Question 17

subset of real numbers

Answer 17

S:= {x∈ℝ: Σ∞ₙ₌₀(aₙ)(xⁿ) converges}
where S!=∅
(given any x you can make a sum which is any real number provided the series converges hence its a subset of the real values)

Question 18

radius of convergence

Answer 18

The largest value of x for which the power series will converge

Question 19

exponential function

Answer 19

The function exp: ℝ-> ℝ is defined by:

exp(x) = Σ∞ₙ₌₀ xⁿ/n!

Question 20

power series about a point

Answer 20

A power series about a point c∈ℝ is an expression of the form:
Σ∞ₙ₌₀(aₙ)(x-c)ⁿ = a₀ + a₁(x-c) + a₂(x-c)² + …

Question 21

power series representation of f on the set

Answer 21

suppose Σ∞ₙ₌₀(aₙ)(x-c)ⁿ converges for |x-c|

Question 22

taylor series

Answer 22

The series Σ∞ₙ₌₀(f⁽ⁿ⁾(x-c)ⁿ)/n! is called the taylor series of f about c and is an example of a power series

Question 23

maclaurin series

Answer 23

in the particular case that c=0, then the taylor series of f is called the maclaurin series