Checklist Flashcards
The completeness axiom
Every set S of real numbers which is bounded above has a supremum B, there exists a real number B s.t sub S = B
Epsolin-Delta definition
For every Epsilon>0 there exists a Delta>0 such that …
If you can keep making N(f(p)) and N(p) smaler
See image on page 128
Definition of continuity of a function at a point (Epsilon-Delta)
lim f(x) = f(x) at any x
There is also always an epsilon for every delta which satisfies the condition of the epsilon-delta definition
Intermediate value theorem for continuous functions
First imagine figure 3.8, with f(x)=k
g(x1)=f(x)-k<0 and g(x2)=f(x)-k>0
Applying bolzanos, we see that g(c)=0 at some point on the intarval [x1,x2].
Extreme value theorem for continious functions
Assume f is continious on a closed intaval [a,b], then we will have a maximum and a minimum in that intaval meaning there wil be some numbers c and d such that
f(c)=sup f and f(d)=inf f
a and b might be the inf and sup, if fx. the graph was a linear function
If (a,b) then there is no specific max or minimum values, it will have sup and inf though
Epsilon - Delta definition of uniform continuity
Use |f(x1) - f(x2)|<Epsilon to make |x1-x2|<Delta appeare. See easy example in “Prove Uniformly Continuous”
There is also another way to prove it, see Trench
Continuity on a closed bounded interval implies uniform continuity
Conceptually: if a set is closed and bounded, then there will always be a delta wich can contain a given epsilon.
Bounded derivative and uniform continuity
If f has a dirivative smaler than a given K>0 then the function is continious
|f’(x)|< K (bigger than or equal to)
Proof by mean value theorem
f(b) - f(a) = f’(c) (b-a)
Definition of differentiability at a point
h’erne går ud med hinanden i gymformlen
Differentiability implies continuity
Remember easy example from book (page 163 - example 7) where we use the identity (gymformel)
f(x+h) = f(x) + h(gymformel)
Mean value theorem for derivatives
The tangent at a given point and the secant between the two endpoints, will at some point be parallel on the graph. Given the Intaval is closed and f is continious.
Use Rolle’s theorem (wiki) on
f(b) - f(a) = f’(c)(b-a)
Definition of a (infinite) sequence (Turing Machine)
A function f with a domain spanining all the positive intergeres, it has a clear starting point, and no specific last term, f(n) is the n’th term of the function.
Turring machine.
Arithmetic infinite sequence - same intaval
Geometric - common ratio, fx multiply by 2
Epsilon - N definition of convergence of a sequence
There exists an epsilon and N so that for every |f(n)-L|<Epsilon>N</Epsilon>
After the N’th value on the x axis, the function can no longer leave the restrictic Epsilon area
Definition of a (infinite) series
See book, very simple, just adding a_n…..
s_1=a_1 s_2=a_1+a_2 and so on
Ad from series, remember the adding together to give 1/2
Definition of convergence of a series
Epsilon - N definition. Imagine the Epsilon band in which the function has to be after the N’th value
A series is the sum of the terms of an infinite sequence of numbers.
If we get closer and closer to a given number
If there exists a number n>N such that |f(n)-L|<Epsilon (L is the sum of all sequences)
Necessary condition for convergence of a series
There are a few tests that can be divided into three main groups 1- “Sufficient Condition” 2-“Necessary Conditions” and “ 3- “Sufficient and Necessary Conditions”
The harmonic series
Diverges
The divergenstest is inconclusive, so we have to use another test (for 1/n fx.) Therefor we use the integral test, integrate 1/n from 1 to a, and then insert infinity when done
Stable bøger
Example
Sigma n=1 goes to infinity of 1/n
Absolute convergense implies convergence
If a sequence converges, then the absolute value of that sequence also converges
Leibniz rule (alternating series test)
If it goes plus minus plus minus and so on, then use The Alternating Series Test to se if it diverges or converges
Intuition: you go forward, and then a little back again and so on, you check for convergens. The positive go forward and negative go backwards
a_n has to be decreasing
a_n has to be positive
lim a_n = 0
If all these are true, then convergens
https://www.youtube.com/watch?v=-lD0skTnqFo
