feat: Add new notes and fix Rolle note

Author: if-loop69420

notes/20240929160535.md

@@ -6,4 +6,4 @@ Respective to a set this means having the following attributes:
 3. Every element in the set has to have an inverse element $a^{-1}$, so that $a*a^{-1}=e$
 4. If the operation on the set is commutative the group is called abelian.
 
-#math #algebraicstructures #group 
+#math #algebraicstructures #algebraicstructures 

notes/20241002214804.md

@@ -3,6 +3,9 @@ A geometric sequence $a_n$ is a series [[20241002211453]], that always grows by
 
 For example $(a_n)_{n>=0}=q^n$ would be a geometric sequence always getting larger by q.
 
-$a_n=2^n=\{1,2,4,8,16,...\}$
+$a_n=2^n=\{1,2,4,8,16,...\}$.
+
+A geometric series always has the following form: 
+$\sum_{n>=0} q^n = 1 + q + q^2 + q^3 +…$
 
 #math #analysis #sequence

notes/20241123132545.md

@@ -1,5 +1,6 @@
 # What is Rolle's theorem? 
-Rolle's theorem states, that if a functions is continuous [[20241110150919]] in $[a,b]$ and differntiable [[20241123130901]]
+Rolle's theorem states, that if a functions is continuous [[20241110150919]] in $[a,b]$ and differntiable [[20241123130901]] and $f(a)=f(b)$.
+
 in $(a,b)$, then there is a point $\xi \in (a,b)$ where $f'(\xi)=0$. 
 
 [[20241123132751]]

notes/20241215165932.md

@@ -3,4 +3,4 @@ Integration can be thought of as two things:
 1. The reverse operation to derivation(differentiation)[[20241123130700]]
 2. A way of calculating the area under a function.
 
-#math #analysis #function #derivatives #integrals
+#math #analysis #functions #derivatives #integrals

notes/20241215170515.md

@@ -2,4 +2,4 @@
 Partial integration refers to applying the following general rule for integration [[20241215170122]]:
 $\int f(x)g'(x)= f(x)g(x)-\int f'(x)g(x)$.
 
-#math #analysis #function #integrals
+#math #analysis #functions #integrals

notes/20250113091546.md

@@ -0,0 +1,4 @@
+# What is the limit of the geometric series?
+The limit of the geometric series[[20241002214804]] (if it converges, which it does if $|q| < 1$) is $\frac{1}{1-q}$.
+
+#math #analysis #series

notes/20250116130550.md

@@ -0,0 +1,6 @@
+# When is a polynomial irreducible? 
+A polynomial $p$ [[20240920124236]] over a field [[20241126093518]] $\mathbb{K}$ is irreducible, if there are no polynomials of degree $≤ deg(p)/2$, that divide it. That means that polynomials of degrees 2 and 3 are only irreducible, if they have no zeroes (in the field $\mathbb{K}$).
+
+A polynomial of degree $deg(p)$ is divisible by a polynomial of degree $deg(p) - m$ if it is divisible by a polynomial of degree $m$.
+
+#math #adm #polynomials #algebraicstructures