Calc Week 1

Question 1

To denote natural numbers

Answer 1

fancy capital N with a line with the first one

Question 2

Sequence and how we denote

Answer 2

a sequence a(n) is a function whose domain is all natural numbers
- an infinite list of number with definite ordering

{a sub n} or (a sub n)

Question 3

a sequence’s terms and how we denote them

Answer 3

a(1), a(2), a(3)….
a sub 1, a sub 2, a sub 3, ….

Question 4

1 - Oscillation Sequence
2- exponential decay sequence
3-Fibonacci sequence
4- harmonic sequence
5 - alternating harmonic sequence

Answer 4

1 - alternates from 1 value to another
{ 4 + (-1) ^ n }

2- gets close to 0 fast
{ 1 / 10^ n}

3- the 2 numbers before it add up to the next

4- {1 / n}

5- alternates from negative to positive
{ (-1)^n / n }

Question 5

null sequence and how to identify one

Answer 5

a sequence {a sub n} converges to zero if
for every epsilon > 0, there exists a positive number m such that
| a sub n| < epsilon
whenever n >= m

for any positive distance epsilon, if you go far enough along the sequence, you will eventually reach a cutoff (m) where the terms of sequence are within epsilon distance of zero

identify one by :
finding its limit
- “plug” in the limit (what has the biggest weight, its reciprocal is multiplied to the top and bottom)
-convert to a function (usually when we would have to use L’H with the sequence because we cannot)
- multiplying by conjugate
(subtracting or adding infinity, multiply time the opposite sign of the same thing on top and bottom)

Question 6

converge

Answer 6

a sub n - L| < epsilon whenever n >= m

{a sub n } converges to L if for every epsilon > 0 there exists a positive number m such that

Question 7

diverges

Answer 7

{ a sub n } diverges to infinity if for every natural number c there is a cutoff m such that a sub n > c when every n >= m

Question 8

limit

Answer 8

if {a sub n} converges to L

L is the limit of {a sub n} as n approaches infinity

Question 9

Finding the limit

Answer 9

• calculation (theorem, just plug in the limit)
  -converting to a function(usually to use L’H rule with it)
• multiplication by conjugate(subtracting or adding infinity, multiply times the same thing (on top and bottom but with opposite sign)

Question 10

algebraic limit theorem

Answer 10

Let {a sub n} and {b sub n} be sequences such that a ub n -> a and b sub n -> b as n -> infinity

then:
lim n -> inf ( a sub n + b sub n) = a + b

lim n -> inf (a sub n - b sub n) = a - b

lim n->inf (a sub n * b sub n) = ab

lim n->inf ( c * a sub n) = ca

lim n->inf ( a sub n/ b sub n) = a/b (as long as we do not get indefinite/undetermined)

Question 11

bounded

Answer 11

there exists real numbers m and M such that
m <= a sub n <= M for all n in natural numbers

Question 12

unbounded

Answer 12

if {a sub n} is not bounded

Question 13

bounded above

Answer 13

if {a sub n} <= M for all n

Question 14

bounded below

Answer 14

if m <= a sub n for all n

Question 15

theorem of convergence then…

Answer 15

let {a sub n} be a convergent sequence.
then {a sub n} is bounded

Question 16

monotonically increasing

Answer 16

{a sub n} is monotonically increasing if
a sub n <= a sub n + 1 for all n

Question 17

monotonically decreasing

Answer 17

if a sub n >= a sub n+1 for all n

Question 18

monotonic

Answer 18

if { a sub n} is either monotonically increasing or monotonically decreasing

Question 19

sequence of partial sums

Answer 19

let {a sub n} be a sequence
the corresponding sequence of partial sums of {a sub n} is the sequence {s sub n}

s sub 1 = a sub 1
s sub 2 = a sub 1 + a sub 2
s sub 3 = a sub 1+ a sub 2 + a sub 3

s sub n = a sub 1 + ….+ a sub n for all n

Question 20

finding if a sequence is monotonic

Answer 20

• look at graph
• compare a sub n+1 - a sub n