Added some idenenties + LHopital
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alexander
@@ -1,6 +1,9 @@
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#import "../lib/common_rewrite.typ" : *
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#import "@preview/mannot:0.3.1"
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#show math.integral: it => math.limits(math.integral)
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#show math.sum: it => math.limits(math.sum)
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#set page(
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paper: "a4",
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margin: (
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@@ -40,40 +43,28 @@
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#columns(4, gutter: 2mm)[
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Allgemeins]
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#grid(
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columns: (auto, auto),
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row-gutter: 2mm,
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column-gutter: 3mm,
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[Dreiecksungleichung], [
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$abs(x + y) <= abs(x) + abs(y)$ \
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$abs(abs(x) - abs(y)) <= abs(x - y)$
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],
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[Cauchy-Schwarz-Ungleichung], [
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$abs(x dot y) <= abs(abs(x) dot abs(y))$
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],
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[Geometrische Summenformel], [
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#MathAlignLeft($ limits(sum)_(k=1)^(n) k = (n(n+1))/2 $)
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],
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[Bernoulli-Ungleichung ], [
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$(1 + a)^n x in RR >= 1 + n a$
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],
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[Binomialkoeffizient], [
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$binom(n, k) = (n!)/(k!(n-k)!)$
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],
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[Binomische Formel], [
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#MathAlignLeft($ (a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $)
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],
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[Fakultäten], [$ 0! = 1! = 1 $],
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[Gausklammer], [
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*Dreiecksungleichung* \
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$abs(x + y) <= abs(x) + abs(y)$ \
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$abs(abs(x) - abs(y)) <= abs(x - y)$ \
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*Cauchy-Schwarz-Ungleichung*\
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$abs(x dot y) <= abs(abs(x) dot abs(y))$ \
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*Geometrische Summenformel*\
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$sum_(k=1)^(n) k = (n(n+1))/2$ \
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*Bernoulli-Ungleichung* \
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$(1 + a)^n x in RR >= 1 + n a$ \
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*Binomialkoeffizient* $binom(n, k) = (n!)/(k!(n-k)!)$
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*Binomische Formel*\
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$(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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*Fakultäten* $0! = 1! = 1$ \
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*Gaußklammer*: \
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$floor(x) = text("floor")(x)$ \
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$ceil(x) = text("ceil")(x)$
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],
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[Bekannte Werte], [
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$ceil(x) = text("ceil")(x)$ \
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*Bekannte Werte* \
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$e approx 2.71828$ ($2 < e < 3$) \
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$pi approx 3.14159$ ($3 < pi < 4$)
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]
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)
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]
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#bgBlock(fill: colorAllgemein)[
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@@ -84,8 +75,20 @@
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#grid(
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columns: (1fr, 1fr),
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row-gutter: 2mm,
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[$ sin(x) = (e^(i x) - e^(-i x))/(2i) $],
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[$ cos(x) = (e^(i x) + e^(-i x))/(2) $]
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[$ cos(x) = (e^(i x) + e^(-i x))/(2) $],
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grid.cell(
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colspan: 2,
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align: center,
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$ tan(x) = 1/2i ln((1+i x)/(1-i x)) $
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),
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grid.cell(
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colspan: 2,
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align: center,
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$ arctan(x) = 1/2i ln((1+i x)/(1-i x)) $
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)
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)
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#subHeading(fill: colorAllgemein)[Trigonmetrie]
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*Additionstheorem* \
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@@ -93,6 +96,10 @@
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$cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \
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$tan(x) + tan(y) = (tan(a) + tan(b))/(1 - tan(a) tan(b))$ \
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$arctan(x) + arctan(y) = arctan((x+y)/(1 - x y))$ \
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$arctan(1/x) + arctan(x) = cases(
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x > 0 : pi/2,
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x < 0 : -pi/2
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)$
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*Doppelwinkel Formel* \
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$cos(2x) = cos^2(x) - sin^2(x)$ \
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@@ -176,8 +183,10 @@
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$forall m,n >= n_epsilon : abs(a_n - a_m) < epsilon$ \
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Cauchyfolge $=>$ Konvergenz)
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- $a_n$ unbeschränkt $=>$ divergenz
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]
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*Konvergent Grenzwert finden*
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Folgen Konvergenz Strategien]
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- Von Bekannten Ausdrücken aufbauen
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- Fixpunk Gleichung: $a = f(a)$ \
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für rekusive $a_(n+1) = f(a_n)$ (Zu erst machen!)
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@@ -189,6 +198,23 @@
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$b_n -> +infinity$: $c_n <= b_n $, wenn $a_n -> +infinity$
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- Zwerlegen in Konvergente Teil folgen \
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(Vorallem bei $(-1)^n dot a_n$)
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*L'Hospital*
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$x in (a,b): limits(lim)_(x->b)f(x)/g(x)$
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(Konvergenz gegen $b$, beliebiges $a$)
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Bendingungen:
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1. $limits(lim)_(x->b)f(x) = limits(lim)_(x->b)g(x)= 0 "oder" infinity$
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2. $g'(x) != 0, x in (a,b)$
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3. $limits(lim)_(x->b) (f'(x))/(g'(x))$ existiert
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$=> limits(lim)_(x->b) (f'(x))/(g'(x)) = limits(lim)_(x->b) (f(x))/(g(x))$
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Kann auch Reksuive angewendet werden!
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]
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#bgBlock(fill: colorFolgen)[
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@@ -212,15 +238,14 @@
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Bekannte Folgen]
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#grid(
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columns: (auto, auto, auto),
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columns: (auto, auto),
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column-gutter: 4mm,
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row-gutter: 2mm,
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align: bottom,
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MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
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[],
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MathAlignLeft($ lim_(n->infinity) k = k, k in RR $),
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grid.cell(colspan: 2, MathAlignLeft($ exp(x) = e^x = lim_(n->infinity) (1 + x/n)^n $)),
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MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $),
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MathAlignLeft($ lim_(n->infinity) k = k, k in RR $),
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MathAlignLeft($ e^x = lim_(n->infinity) (1 + x/n)^n $),
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grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) q^n = cases(
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0 &abs(q),
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1 &q = 1,
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@@ -246,8 +271,6 @@
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- *Absolute Konvergenz* \
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$limits(sum)_(n=1)^infinity abs(a_n) = a => limits(sum)_(n=1)^infinity a_n$ konvergent
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- *Partialsummen* \
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ALLE Partialsummen von $limits(sum)_(k=1)^infinity abs(a)$ beschränkt\
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$=>$ _Absolute Konvergent_
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@@ -278,19 +301,6 @@
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2. $rho = lim_(n -> infinity) root(n, abs(a_(n+1))) $ \
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divergent: $rho > 1$, keine Aussage $rho = 1$, konvergent $rho < 1$
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- *Geometrische Reihe*
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$limits(sum)_(n=0)^infinity q^n$
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- konvergent $abs(q) < 1$, divergent $abs(q) >= 1$
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- Grenzwert: (Muss $n=0$) $=1/(1-q)$
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- *Harmonische Reihe* $limits(sum)_(n=0)^infinity 1/n = +infinity$
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- *Reihendarstellungen*
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1. $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
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2. $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
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3. $sin(x) = limits(sum)_(n=0)^infinity $
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4. $cos(x) = limits(sum)_(n=0)^infinity $
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]
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#bgBlock(fill: colorReihen)[
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@@ -305,9 +315,16 @@
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*Harmonische Reihe:* $sum_(n=0)^infinity 1/n = +infinity$
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*Andere*
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- $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
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- $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
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*Reihendarstellungen*
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#grid(
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columns: (1fr, 1fr),
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gutter: 3mm,
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row-gutter: 2mm,
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$e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$,
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$ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$,
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$sin(x) = limits(sum)_(n=0)^infinity $,
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$cos(x) = limits(sum)_(n=0)^infinity $
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)
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]
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#colbreak()
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@@ -421,7 +438,7 @@
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{ color.hsl(180deg, 81.82%, 95.69%) },
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[$1/(q + x) x^(q+1)$], [$x^q$], [$q x^(q-1)$],
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[$ln abs(x)$], [$1/x$], [$-1/x^2$],
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[$x ln(a x) - x$], [$ln(a x)$], [$1 / x$],
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[$x ln(a x) - x$], [$ln(a x)$], [$a / x$],
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[$2/3 sqrt(a x^3)$], [$sqrt(a x)$], [$a/(2 sqrt(a x))$],
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[$e^x$], [$e^x$], [$e^x$],
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[$a^x/ln(a)$], [$a^x$], [$a^x ln(a)$],
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@@ -470,6 +487,50 @@
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3. $x$-kürzen sich weg
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])
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#bgBlock(fill: colorIntegral, [
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#subHeading(fill: colorIntegral, [Integral])
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*Riemann Integral*\
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$limits(sum)_(x=a)^(b) f(i)(x_())$
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Summen: $integral f(x) + g(x) d x = integral f(x) d x + integral g(x)$
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Vorfaktoren: $integral lambda f(x) d x = lambda f(x) d x$
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*Integral Type*\
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- Eigentliches Int.: $integral_a^b f(x) d x$
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- Uneigentliches Int.: \
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$limits(lim)_(epsilon -> 0) integral_a^(b + epsilon) f(x) d x$ \
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$limits(lim)_(epsilon -> plus.minus infinity) integral_a^(epsilon) f(x) d x$
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- Unbestimmtes Int.: $integral f(x) d x = F(x) + c, c in RR$- Uneigentliches Int.:
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*Cauchy-Hauptwert*
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$integral_(-infinity)^(+infinity) f(x)$ \
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NUR konvergent wenn: \
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$limits(lim)_(R -> -infinity) integral_(R)^(a) f(x) d x$ und $limits(lim)_(R -> infinity) integral_(a)^(R) f(x) d x$ konvergent für $a in RR$
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$integral_(-infinity)^(infinity) f(x) d x$ existiert \
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$=> lim_(M -> infinity) integral_(-M)^(M) f(x) d x = integral_(-infinity)^(infinity) f(x) d x$
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*Partial Integration*
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$integral u(x) dot v'(x) d x = u(x)v(x) - integral u'(x) dot v(x)$
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*Subsitution*
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$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot 1/(g'(x)) d x$
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1. Ersetzung: $ d x := d t dot g'(x)$ und $t := g(x)$
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2. Grenzen: $t_0 = g(x_0)$, $t_1 = g(x_1)$
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3. $x$-kürzen sich weg
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*Absolute "Konvergenz"* \
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Wenn $g(x)$ konvergent,
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$abs(f(x)) <= g(x) => $ $f(x)$ konvergent
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])
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]
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#bgBlock(fill: colorAllgemein, [
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@@ -524,23 +585,4 @@ Konvergenz Radius $R = [0, infinity)$$$
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)$
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$ R = limsup_(n -> infinity) $
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#bgBlock(fill: colorIntegral, [
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#subHeading(fill: colorIntegral, [Integral])
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Summen: $integral f(x) + g(x) d x = integral f(x) d x + integral g(x)$
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Vorfaktoren: $integral lambda f(x) d x = lambda f(x) d x$
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*Partial Integration*
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$integral u(x) dot v'(x) d x = u(x)v(x) - integral u'(x) dot v(x)$
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*Subsitution*
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$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot g'(x) d x$
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1. Ersetzung: $ d x := d t dot 1/(g'(x))$ und $t := g(x)$
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2. Grenzen: $t_0 = g(x_0)$, $t_1 = g(x_1)$
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3. $x$-kürzen sich weg
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])
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