midterm 1 8.1-8.3

sequences

an infinitely long list of numbers

If you start a sequence a_n and want to take the negative of every other term, then

take the sequence (-1)ⁿa_n or (-1)ⁿ⁺¹a_n

if the difference between adjacent terms in the sequence is a constant c, then the formula is

c*n+b

examples of c*n+b: {2, 5, 8, 11,…}

a_n=3n-1 ; 3 is being added every time so c is 3 and b is -1 bc when you substitute 3 in you get -1

if the distances between distances is a constant then the formula for the sequence is

an²+bn+d

example of an²+bn+d: {2, 5, 10, 17, …}

n²+1

if the terms in a sequence one and only one number L, we say the sequence

a_n converges to L and write lim n—>inf a_n=L ; if not we say a_n diverges

ex: the sequence {1/2, 1/4, 1/8, 1/16, …}

converges to 0

ex: the sequence {9, 8.1, 8.01, 8.001, 8.0001, …}

converges to 8

ex: the sequence {-1, 1, -1, 1, -1, …}

diverges (bc it oscillates)

to figure out of a_n converges you can

take the limit and multiply by the reciprocal of the biggest exponent (which will make all the smaller ones 0) and get where it diverges to

if we have a function f(x) with domain R

we can make a sequence from it {f(1), f(2), f(3),…}

Fact: lim n—>inf f(n) = lim x—> inf f(x)

if this limit exists

arctan’s limit is

pi/2

a sequence a_n is bounded if there is…

a number C where a_n always is in [-c,c] no matter what n is

ex: {1, 1/2, 1/3, 1/4,…}

is bounded

ex: {1, 2, 3, 4, ..}

is not bounded

ex: {-1, -2, -3, -4, …}

is not bounded

ex: {0, 1, 0, 2, 0, 3, 0, 4, …}

is not bounded

ex: {-1, 1, -1, 1, -1, …}

is bounded

if the limit is inf or -inf

it CANNOT be bounded

If a sequence converges, it is

bounded

If a sequence is bounded then it

does NOT have to converge

a sequence is monotonically increasing if

a_n+1>=a_n for all n , i.e. every tetm is larger or equal to one before it

every bounded monotonically increasing sequence

converges

if you want to compute a limit of lim x—> ? f(x)/g(x) for some functions and direct sub gives 0/0 or +- inf/inf then

lim x—>? f(x)/g(x) = lim x—>? f’(x)/g’(x) (L’Hopitals rule)

if f(x) and g(x) are functions, we say g

grows faster than f if lim x—→inf f(x)/g(x) = 0

a^x always

grows faster than any polynomial if a>1

the infinite sum (or series) Summation n=1 to infinity a_n means

the limit of {a₁,a₂,a₃,…} in other words lim n—> inf summation of n=1 to infinity a_n.

the summation a_n

cannot converge AND lim n—>inf a_n =/ 0

If lim n—> inf a_n=/ 0

then summation n=1 to inf a_n diverges

if summation n=1 to inf a_n converges

then lim n—> inf a_n =0

denominator > numerator

converges to 0

denominator = numerator

converges to coefficients

denominator < numerator

diverges

direct comparison test: given numbers a_n>=b_n>= 0

1) if summation n=1 to inf b_n diverges, then summation of a_n diverges

2) if summation a_n converges, then summation b_n converges

p-series

summation 1/n^p converges if p>1 and diverges if p<= 1

integral test: if you take a function,

f(1) + f(2) + f(3) +…. converges <=> integral of 1 to inf f(x) dx converges

geometric series

series of the form a+ar+ar²+ar³+ar^4+…. summation n=0 to inf a*rⁿ

S_n

a/(1-r) if r is in (-1,1) and diverges

Telescoping

when you put the terms and can cancel them out

Given a sequence of numbers a₁, a₂, a_{3, …}

we have said the N^th partial sum of this sequence is S_n summation

speed tiers

ln (n) > polynomials > exponentials (base>1) > factorials > nⁿ

true or false: all monotonically increasing sequences converge

false (an = n)

true or false: a sequence is bounded, it must converge

false an = (-1)^n

true or false: if a sequence converges it must be bounded

true (always true)