calculus

theorem: radical 2 is irrational

proof?

proof

definition of upper bound

let A c R, we say that a number a is an upper bound of set A if:

x</= a for each x belonging to A

definition of bounded above

we say that the set A is bounded above if it admits an upper bound

axiom of continuity (supremum)

every non empty subset A c R with an upper bound has the least upper bound (supremum) in R

definition supremum

we call supremum of A (SupA) the least upper bound.

The SupA is also MaxA if the number belongs to the set, otherwise the set has no max but just the sup.

if the set has no upper bound, SupA =\infty

axiom of continuity (infimum)

\forall A\ne\phi \exists InfA

InfA = greatest lower bound in R

theorem: 0. 9 periodic = 1

proof?

we write 0.\overline{9}=0+\frac{9}{10}+\frac{9}{100}+....+\frac{9}{10^{n}}=\frac{9}{10}\Sigma\left(\frac{1}{10^{k}}\right) using geometric series formula we can say

\Sigma\left(\frac{1}{10^{k}}\right)=\frac{1}{1-\frac{1}{10}}\cdot\frac{9}{10}=1

definition of complex numbers

a complex number is a polynomial of degree \le1 in the unknown i

a+ bi

definition of conjugate of complex number

let z=x+iy be a complex number. Then we call conjugate of z (\overline{z} ) the reflection of z in the real axis: \overline{z}\colon=x-iy

definition of module of complex number z=x+iy

let z be a complex number. then we call module of z (\left\vert z\right\vert ) the distance of z from the origin: \left\vert z\right\vert=\sqrt{x^2+y^2}\in R

polar/trigonometric form of a complex number

knowing that z= a + bi

we say that a=r cos \theta

and b= r sin \theta

so z=r\left(\cos\theta+i\sin\theta\right)

where r is the module of z

fundamental theorem of algebra

every polynomial of order n admits exactly n roots in C

definition of sequence

a sequence {an} is a rule that assigns to every n € N a real value an

N\ni n\rightarrow an\in R

definition of bounded above for a sequence

we say that a sequence {an} is bounded above if \exists M>0,M\in R\vert an\le M\forall n\in N

definition of bounded below sequence

we say that a sequence {an} is bounded below if \exists m>0\vert m\le an\forall n\in N

definition of bounded sequence

we say that {an} is bounded if it’s bounded above and below

definition of monotone sequence

we say that a sequence is monotone or monotonic if it’s either increasing or decreasing

definition of non decreasing sequence

we say that {an} is monotonic non decreasing if a_{n+1}\ge a_{n}\forall n

definition of strictly increasing sequence

we say that {an} is monotonic strictly increasing if an+1>an for every n

definition of non increasing sequence

we say that {an} is monotonic non increasing if a_{n+1}\le an\forall n

definition of decreasing sequence

we say that {an} is monotonic strictly decreasing if a_{n+1}<an\forall n

definition of finite limit for a sequence

let {an} be a sequence, L € R. we say that {an} converges to L and write \lim_{n\rightarrow\infty}\left(an\right)=L if \forall\char"0190 >0\exists N=N\char"0190 >0\vert\left\vert an-L\right\vert<\char"0190 \forall n\in N

definition of limit of a sequence going to + infinity+\infty

let {an} be a sequence. we say that {an} goes to infinity and we write \lim_{n\rightarrow\infty}\left(an\right)=+\infty if \forall M>0\exists N=N_{M}>0\vert an>M\forall n>N

definition of limit of a sequence going to - infinity

let {an} be a sequence. we say that {an} goes to -\infty and we write \lim_{n\rightarrow\infty}\left(an\right)=-\infty if ?????

definition of regular sequence

let {an} be a sequence. we say that {an} is regular if it has a limit

definition of convergent sequence

we say that a regular sequence is convergent if the limit of the sequence as n goes to infinity, goes to L with L€R

definition of divergent sequence

we say that a regular sequence is divergent if the limit of the sequence exists but is infinite,\lim_{n\rightarrow\infty}\left(an\right)=\pm\infty

definition of irregular sequence

we say that {an} is irregular if it’s neither convergent nor divergent