Lösungen / Solutions/I1S2Ue/ExT3

L = Lektion/ Leçon   -- Sprechstunde / heures de consultation
W = Work / Arbeit / Travail
A  =Anhang/  Annexe / Appendice

Ueb/Ex 1      W

a

Sum[
      5*(1/4)^(k + 1)  + 6*(1/3)^(k - 1), {k, 1, 
        Infinity}]

Konv. (geom. Reihen!) /  Conv . (Séries géom.!)

b

Sum[(k + k^3)/(1 - k + k^2 - k^3 + k^4 - k^5) , {k, 1, 
        Infinity}]

Konv. (p-Reihen!) /  Conv . (Séries "p"!)

c

Sum[(-1)^k  1/(k + 7) , {k, 1, Infinity}]

Konv. (Leibniz!) /  Conv . (Leibniz!)

d

Sum[(-1)^k  k/(k + 7) , {k, 1, Infinity}]

Nicht konv. (Leibniz!) /  Ne pas conv . (Leibniz!)

e

?Apart

Apart[1/(k^2 + 2  k - 3)]

Sum[(-1)^k 1/(k^2 + 2  k - 3) , {k, 2, 
        Infinity}]

Sum[ 1/(k^2 + 2  k - 3) , {k, 2, Infinity}]

f

Sum[(10  x)^k /k, {k, 1, Infinity}]

((10  x)^k /k)^(1/k) == 10  x/k^(1/k)

Limit[k^(1/k), k -> Infinity]

10x  <  1

<< Algebra`InequalitySolve`

InequalitySolve[Abs[10  x] < 1, x]

Konv., Wurzelkriterium /  Conv., Cauchy  ==>  -(1/10) < x < (1/10)

D[(10  x)^k /k, x]

Sum[(10^k) (x^(-1 + k)), {k, 1, Infinity}]

Konv., geom. Reiche /  Conv.,série géom.

Integrate[-(10/(-1 + 10 x)), x]

-1+10 x   ===>  Abs[-1+10 x],   -(1/10) < x < (1/10)  ===> Resultat  -Log[1-10 x]

-Log[-1 + 10 *2/10]

Integrate[-(10/(-1 + 10 t)), {t, 2/10, x}]

-(1/10) < x < (1/10)

Sum[(10* 1/100)^k /k, {k, 1, Infinity}]

Integrate[-(10/(-1 + 10 t)), {t, 1, 1 + 1/10}]

Integrate[-(10/(-1 + 10 t)), {t, 1 + 1/10, 1}]

g

(1 + k) (2 + 3  k) // Expand

Sum[(1 + k) (2 + 3  k), {k, 1, n}]

Sum[(1 + k) (2 + 3  k), {k, 1, n}] // Expand

Ueb/Ex 2      W

a    (Subscript[a, n])(x) = ?

Series[Sin[x],  {x,  0,  20}]

Series[Sin[x]/x,  {x,  0,  20}]

Series[Sin[x]/x,  {x,  0,  20}] // Normal

Normal[Series[Sin[x],  {x,  0,  20}]]/x   //  Expand

a[x_, 
          n_] := (((-1))^((n)))*(1/((2  n + 1)!)) *(((x))^(2  n));
a[x, 0]

a[x, 1]

a[x, 2]

Series[(Sin[x]/x  - 1)/x^2,  {x,  0,  20}] // Normal

Solve[(Sin[x]/x  - 1)/x^2 == -1/6, {x}]

b Konvergenzradius / Rayon de convergence

Quotientenkriterium / Critère de d'Alembert

a[x, n + 1]/a[x, n] // Simplify

a[x, n + 1]/
        a[x, n] /.  ((1 + 2 n)!/(1 + 
                  2 (1 + n))!) -> 
        1/((3 + 2  n) (2 + 2  n))

-((x^(-2 n + 2 (1 + n)))/((2 + 2 n) (3 + 
                  2 n)))  // Simplify

r00[x_] := 
      Limit[-((x^2)/(6 + 10 n + 4 (n^2))), n -> Infinity]; 
    r00[x]

===>    |0| = q < 1       für Konv. / pour la conv.

Integrate[1-((x^2)/6)+((x^4)/120)-((x^6)/5040)+((x^8)/362880)-((x^10)/39916800)+((x^12)/6227020800)-((x^14)/1307674368000)+((x^16)/355687428096000)-((x^18)/121645100408832000)+((x^20)/51090942171709440000)+O[x]^21,x]

Integrate[
      1 - ((x^2)/6) + ((x^4)/120) - ((x^6)/5040) + ((x^8)/362880) - ((x^10)/39916800) + 
        ((x^12)/6227020800) - ((x^14)/1307674368000) + ((x^16)/355687428096000) - 
        ((x^18)/121645100408832000) + ((x^20)/51090942171709440000), 
      x]

Integrate[
      1 - ((x^2)/6) + ((x^4)/120) - ((x^6)/5040) + ((x^8)/362880) - ((x^10)/39916800) + 
        ((x^12)/6227020800) - ((x^14)/1307674368000) + ((x^16)/355687428096000) - 
        ((x^18)/121645100408832000) + ((x^20)/51090942171709440000), {x, 0, 
        t}]

Integrate[Sin[x]/x, {x, 0, t}]

Plot[{Sin[t]/t, 
          t - ((t^3)/18) + ((t^5)/600) - ((t^7)/35280) + ((t^9)/3265920) - 
            ((t^11)/439084800) + ((t^13)/80951270400) - ((t^15)/19615115520000) + 
            ((t^17)/6046686277632000) - ((t^19)/2311256907767808000) + 
            ((t^21)/1072909785605898240000), 
          1 - ((t^2)/6) + ((t^4)/120) - ((t^6)/5040) + ((t^8)/362880) - 
            ((t^10)/39916800) + ((t^12)/6227020800) - ((t^14)/1307674368000) + 
            ((t^16)/355687428096000) - ((t^18)/121645100408832000) + 
            ((t^20)/51090942171709440000)
}, {t, -10, 
          10}];

Ueb/Ex 3      W

Sin[x + Pi/4] // TrigExpand

Plot[{Sin[x], Cos[x], Sin[x] + Cos[x], Sin[x + Pi/4]}, {x, 0, 
          2  Pi}];

Plot[{Sin[x] + Cos[x], Sin[x + Pi/4], Sqrt[2] Sin[x + Pi/4]}, {x, 0, 
          2  Pi}];

Series[Sin[x], {x, 0, 10}]

Series[Cos[x], {x, 0, 10}]

Series[Sin[x], {x, 0, 10}] + Series[Cos[x], {x, 0, 10}]

Series[Sqrt[2] Sin[x + Pi/4], {x, 0, 10}]

Ueb/Ex 4      W

Solve[(Sin[x]/x  - 1)/x^2 == -1/6, {x}]

Normal[Series[Sin[x]/x,  {x,  0,  20}]] - 1

(Normal[Series[Sin[x]/x,  {x,  0,  20}]] - 1)/x^2 // 
      Simplify

Simplify[(Normal[Series[Sin[x]/x,  {x,  0,  20}]] - 1)/
          x^2] == -1/6

((x^2)/120) - ((x^4)/5040) + ((x^6)/362880) - ((x^8)/39916800) + 
        ((x^10)/6227020800) - ((x^12)/1307674368000) + ((x^14)/355687428096000) - 
        ((x^16)/121645100408832000) + ((x^18)/51090942171709440000) == 
      0

Solve[(Normal[Series[Sin[x]/x,  {x,  0,  20}]]  - 1)/
          x^2 == -1/6, {x}]