Finally added potenzreihen
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alexander
@@ -3,6 +3,7 @@
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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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#let lim = $limits("lim")$
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#set text(7pt)
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@@ -46,27 +47,48 @@
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Allgemeins]
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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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*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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#grid(
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columns: (1fr, 1fr),
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inset: 0mm,
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gutter: 2mm,
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[
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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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[
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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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[
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*Geometrische Summenformel*\
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$sum_(k=1)^(n) k = (n(n+1))/2$ \
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],
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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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[
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*Binomialkoeffizient* $binom(n, k) = (n!)/(k!(n-k)!)$
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],
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[
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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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],
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[
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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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*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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[
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*Fakultäten* $0! = 1! = 1$ \
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],
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)
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]
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#bgBlock(fill: colorAllgemein)[
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@@ -81,12 +103,12 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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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) $],
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grid.cell(
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colspan: 2,
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colspan: 1,
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align: center,
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$ 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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colspan: 1,
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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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@@ -96,7 +118,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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*Additionstheorem* \
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$sin(x+y) = cos(x)sin(y) + sin(x)cos(y)$ \
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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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$tan(x +y) = (tan(a) + tan(b))/(1 - tan(a) tan(b))$ \
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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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@@ -108,19 +130,13 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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$sin(2x) = 2sin(x)cos(x)$
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#grid(
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gutter: 5mm,
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columns: (auto, auto),
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[$cos^2(x) = (1 + cos(2x))/2$],
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[$sin^2(x) = (1 - cos(2x))/2$]
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)
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$cos^2(x) + sin^2(x) = 1$
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git config pull.rebase falsegit config pull.rebase false
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#grid(
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gutter: 5mm,
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columns: (auto, auto),
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[$cos(-x) = cos(x)$],
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[$sin(-x) = -sin(x)$],
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gutter: 2mm,
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columns: (auto, auto, auto),
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$cos^2(x) = (1 + cos(2x))/2$,
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$sin^2(x) = (1 - cos(2x))/2$,
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$cos(-x) = cos(x)$,
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$sin(-x) = -sin(x)$,
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grid.cell(colspan: 2, $cos^2(x) + sin^2(x) = 1$)
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)
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Subsitution mit Hilfsvariable
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@@ -134,7 +150,8 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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[$tan(x)=-cot(x + pi/2)$],
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[$cot(x)=-tan(x + pi/2)$],
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[$cos(x - pi/2) = sin(x)$],
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[$sin(x + pi/2) = cos(x)$],
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[$sin(x + pi/2) = cos(x)$],
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)
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$sin(x)cos(y) = 1/2sin(x - y) + 1/2sin(x + y)$
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@@ -143,9 +160,9 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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$arccos(x) = -arcsin(x) + pi/2 in [0, pi]$
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]
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// Folgen Allgemein
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Folgen]
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$ lim_(x -> infinity) a_n $
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*Beschränkt:* $exists k in RR$ sodass $abs(a_n) <= k$
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- Beweiße: durch Induktion
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@@ -155,17 +172,14 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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*Monoton fallend/steigended*
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- Beweise: Induktion
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#grid(columns: (1fr, 1fr),
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gutter: 1mm,
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row-gutter: 2mm,
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inset: 0.2mm,
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align(top+center, [*Fallend*]), align(top+center, [*Steigend*]),
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[$ a_(n+1) <= a_(n) $],
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[$ a_(n+1) >= a_(n) $],
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[$ a_(n+1)/a_(n) < 1 $],
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[$ a_(n+1)/a_(n) > 1 $],
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[$ a_(n+1) <= a_(n), quad a_(n+1) >= a_(n) $],
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[$ a_(n+1)/a_(n) < 1, quad a_(n+1)/a_(n) > 1 $],
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)
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*Konvergentz Allgemein*
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$ lim_(n -> infinity) a_n = a $
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$lim_(n -> infinity) a_n = a$
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$forall epsilon > 0 space exists n_epsilon in NN$ sodass \
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- Konvergent $-> a$: $a_n in [a - epsilon, a + epsilon] $
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@@ -177,31 +191,38 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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*Konvergentz Häufungspunkte*
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- $a_n -> a <=>$ Alle Teilfolgen $-> a$
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*Konvergenz Beweißen*
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- Monoton UND Beschränkt $=>$ Konvergenz
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NICHT Umgekehert
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- (Cauchyfolge \
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$forall epsilon > 0 space exists n_epsilon in NN space$ sodass \
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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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*Folgen in $CC$* (Alle Regeln von $RR$ gelten)\
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- $z_n in CC : lim z_n <=> lim abs(z_n) = 0$
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- Zerlegen in $a + b i$ oder $abs(z) dot e^(i phi)$
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]
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// Folgen Strat
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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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- *Monoton UND Beschränkt $=>$ Konvergenz*
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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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- Bernoulli-Ungleichung Folgen der Art $(a_n)^n$: \
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$(1 + a)^n >= 1 + n a$
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- Sandwitchtheorem:\
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$b_n -> x$: $a_n <= b_n <= c_n$, wenn $a_n -> x$ und $c_n -> x$ \
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$b_n -> -infinity$: $b_n <= c_n$, wenn $c_n -> -infinity$ \
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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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- (Cauchyfolge \
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$forall epsilon > 0 space exists n_epsilon in NN space$ sodass \
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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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*Divergenz*
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- $a_n$ unbeschränkt $=>$ divergenz
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- Vergleichskriterium: \
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$b_n -> -infinity$: $b_n <= c_n$, wenn $c_n -> -infinity$ \
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$b_n -> +infinity$: $c_n <= b_n $, wenn $a_n -> +infinity$
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]
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// L'Hospital
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[L'Hospital]
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$x in (a,b): limits(lim)_(x->b)f(x)/g(x)$
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@@ -211,7 +232,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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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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3. $limits(lim)_(x->b) (f'(x))/(g'(x))$ konveriert
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$=> limits(lim)_(x->b) (f'(x))/(g'(x)) = limits(lim)_(x->b) (f(x))/(g(x))$
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@@ -220,8 +241,9 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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Bei "$infinity dot 0$" mit $f(x)g(x) = f(x)/(1/g(x))$
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]
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// Bekannte Folgen
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Konvergent Folge Regeln]
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#subHeading(fill: colorFolgen)[Bekannte Folgen]
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#grid(
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columns: (auto, auto),
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align: bottom,
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@@ -236,10 +258,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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MathAlignLeft($ lim_(n->infinity) abs(a_n) = abs(a) $),
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MathAlignLeft($ lim_(n->infinity) c dot a_n = c dot lim_(n->infinity) a_n $),
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)
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]
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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),
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column-gutter: 4mm,
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@@ -258,6 +277,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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)
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]
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// Teilfolgen
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Teilfolgen]
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$ a_k subset a_n space (text("z.B") k= 2n + 1) $
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@@ -267,6 +287,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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- Wenn alle $a_k$ gegen #underline([genau eine]) Häufungspunk konverigiert $<=> a_n$ konvergent
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]
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// Reihen
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#bgBlock(fill: colorReihen)[
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#subHeading(fill: colorReihen)[Reihen]
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$limits(lim)_(n->infinity) a_n != 0 => limits(sum)_(n=1)^infinity a_n$ konverigiert NICHT \
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@@ -304,12 +325,34 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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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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*Reihen in $CC$*
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- Alles
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]
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#bgBlock(fill: colorReihen)[
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#subHeading(fill: colorReihen)[Potenzreihen]
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$P(z) = sum_(n=0)^infinity a_n dot (z- z_0)^n quad z,z_0 in CC$
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#grid(
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columns: (auto, auto),
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column-gutter: 5mm,
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row-gutter: 1.5mm,
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[*Konvergenzradius*], [$|z - z_0| < R : $ absolute Konvergenz],
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[], [$|z - z_0| = R : $ Keine Aussage],
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[], [$|z - z_0| > R : $ Divergent]
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)
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#grid(
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columns: (1fr, 1fr),
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$R = lim_(n->infinity) abs(a_n/(a_(n+1))) = 1/(lim_(n->infinity) root(n, abs(a_n)))$,
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$R = limits(liminf)_(n->infinity) abs(a_n/(a_(n+1))) = 1/(limits(limsup)_(n->infinity) root(n, abs(a_n)))$
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)
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|
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]
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|
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// Bekannte Reihen
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#bgBlock(fill: colorReihen)[
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#subHeading(fill: colorReihen)[Bekannte Reihen]
|
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*Geometrische Reihe:* $sum_(n=0)^infinity q^n$
|
||||
@@ -317,19 +360,22 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
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- Grenzwert: (Muss $n=0$) $=1/(1-q)$
|
||||
|
||||
*Harmonische Reihe:* $sum_(n=0)^infinity 1/n = +infinity$
|
||||
|
||||
*Binomische Reihe:*
|
||||
|
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*Reihendarstellungen*
|
||||
#grid(
|
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columns: (1fr, 1fr),
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||||
gutter: 3mm,
|
||||
row-gutter: 2mm,
|
||||
row-gutter: 3mm,
|
||||
$e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$,
|
||||
$ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$,
|
||||
$sin(x) = limits(sum)_(n=0)^infinity $,
|
||||
$cos(x) = limits(sum)_(n=0)^infinity $
|
||||
$sin(x) = limits(sum)_(n=0)^infinity (-1)^n (z^(2n+1))/((2n + 1)!)$,
|
||||
$cos(x) = limits(sum)_(n=0)^infinity (-1)^n (z^(2n))/((2n)!)$
|
||||
)
|
||||
]
|
||||
|
||||
// Ableitung
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
#subHeading(fill: colorAbleitung)[Funktionen]
|
||||
|
||||
@@ -346,7 +392,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
injektiv UND Surjectiv $<=>$ Umkehrbar
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
// Funktions Sätze
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
#subHeading(fill: colorAbleitung)[Funktionen Sätze]
|
||||
$f(x)$ diff'bar $=> f(x)$ stetig
|
||||
@@ -367,7 +413,8 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
$=> exists x_0 : f'(x_0)=(f(b) - f(a))/(a-b)$
|
||||
|
||||
- *Mittelwertsatz der Integralrechnung*\
|
||||
|
||||
$g -> RR "integrierbar," g(x)>= 0 forall x in [a,b]$\
|
||||
$exists xi in [a,b] : integral_a^b f(x)g(x) d x = f(xi) integral_a^b g(x) d x$
|
||||
|
||||
- *Satze von Rolle* \
|
||||
diffbar $x in (a,b)$\
|
||||
@@ -378,6 +425,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
|
||||
]
|
||||
|
||||
// Stetigkeit
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
#subHeading(fill: colorAbleitung)[Stetigkeit]
|
||||
*Allgemein*
|
||||
@@ -419,6 +467,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
)
|
||||
]
|
||||
|
||||
// Ableitung
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
#subHeading(fill: colorAbleitung)[Ableitung]
|
||||
*Differenzierbarkeit*
|
||||
@@ -453,6 +502,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
- Kettenregel: $f(g(x)) : f'(g(x)) dot g'(x)$
|
||||
],
|
||||
|
||||
// Ableitungstabelle
|
||||
#block([
|
||||
#set text(size: 10pt)
|
||||
#table(
|
||||
@@ -492,7 +542,7 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
[$op("artanH")(x)$], [$1/(1 - x^2)$],
|
||||
)
|
||||
])
|
||||
#colbreak()
|
||||
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
#subHeading(fill: colorAbleitung)[Extremstellen, Krümmung, Monotonie]
|
||||
|
||||
@@ -560,7 +610,6 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
Notwendig $f''(x) lt.gt 0$
|
||||
]
|
||||
|
||||
|
||||
#bgBlock(fill: colorIntegral, [
|
||||
#subHeading(fill: colorIntegral, [Integral])
|
||||
|
||||
@@ -614,7 +663,6 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
$limits(lim)_(epsilon -> plus.minus infinity) integral_a^(epsilon) f(x) d x$
|
||||
- Unbestimmtes Int.: $integral f(x) d x = F(x) + c, c in RR$- Uneigentliches Int.:
|
||||
|
||||
|
||||
*Cauchy-Hauptwert*
|
||||
|
||||
$integral_(-infinity)^(+infinity) f(x)$ \
|
||||
@@ -662,53 +710,3 @@ $(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
|
||||
)
|
||||
])
|
||||
]
|
||||
|
||||
|
||||
#pagebreak()
|
||||
|
||||
== Folgen in $CC$
|
||||
|
||||
$z_n in C: lim z_n <=> lim abs(z_n -> infinity) = 0$
|
||||
|
||||
Alle folgen regelen gelten
|
||||
|
||||
Complexe Folge kann man in Realteil und Imag zerlegen
|
||||
|
||||
z.B.
|
||||
|
||||
$z_n = z^n z in CC$
|
||||
|
||||
$z = abs(z) dot e^(i phi) = abs(z)^n$
|
||||
|
||||
== Reihen in $CC$
|
||||
|
||||
Fast alles gilt auch.
|
||||
|
||||
Bis auf Leibnitzkriterium weil es keine Monotonie gibt
|
||||
|
||||
Geometrische Reihe gilt.
|
||||
|
||||
Exponential funktion
|
||||
|
||||
#MathAlignLeft($ e^z = lim_(n -> infinity) (1 + z/n)^n = sum_(n=0)^infinity (z^n)/(n!) space z in CC $)
|
||||
|
||||
Vorsicht: $(b^a)^n = b^(a dot c)$
|
||||
|
||||
Potenzreihen: Eine Fn der form:
|
||||
|
||||
#MathAlignLeft($ P(z) = sum^(infinity)_(n=0) a_n dot (z - z_0)^n space z, z_0 in CC $)
|
||||
|
||||
=== Satz
|
||||
|
||||
Konvergenz Radius $R = [0, infinity)$$$
|
||||
|
||||
1. $R = 0$ Konvergiet nur bei $z = 0$
|
||||
|
||||
2. $R in R : cases(
|
||||
z in CC &abs(z - z_0) < R &: "abs Konvergent",
|
||||
z in CC &abs(z - z_0) = R &: "keine Ahnung",
|
||||
z in CC &abs(z - z_0) > R &: "Divergent"
|
||||
)$
|
||||
|
||||
$ R = limsup_(n -> infinity) $
|
||||
|
||||
|
||||
Reference in New Issue
Block a user