Demandes associées au chapitre-notebook: "Suites et Séries"

Pour accéder au code relatif à un exemple, double-cliquer sur le triangle du crochet icônique à droite.

Exemple 1

u[n_] := (10/3) u[n - 1] - u[n - 2]

u[0] = 1; u[1] = (1/3);

u[2]

u[3]

Table[u[n], {n, 1, 10}]

Timing[Table[u[n], {n, 1, 10}]]

?u

Exemple 2

v[n_] := v[n] = (10/3) v[n - 1] - v[n - 2]

v[0] = 1; v[1] = (1/3);

Timing[Table[v[n], {n, 1, 10}]]

?v

TableForm[Table[N[v[n]], {n, 1, 20}]]

ListPlot[%];

ListPlot[%%, PlotJoined -> True];

Exemple 3

Plot[
      Evaluate[Table[Sin[n x]/(n (Sqrt[x])), {n, 1, 3}]], {x, (π/4), 
        π}];

Plot[
      Evaluate[Table[Sin[n x]/(n (Sqrt[x])), {n, 1, 4}]], {x, (π/4), 
        π}];

Plot[
      Evaluate[Table[Sin[n x]/(n (Sqrt[x])), {n, 1, 10}]], {x, (π/4), 
        π}];

Plot[
      Evaluate[Table[Sin[n x]/(n (Sqrt[x])), {n, 1, 20}]], {x, (π/4), 
        π}];

Clear[f]

Exemple 4

Plot[
      Evaluate[Table[
          ((1 - (r^2))/(1 - 2 r Cos[2 π t] + (r^2))), {r, 0.1, 0.6, 
            0.1}]], {t, -0.5, 0.5}];

Exemple 5

Plot[
      Evaluate[Table[
          ((2 ((1 - (r^2)) Sin[2 π t]^2))/(((
                1 - 2 r Cos[2 π t] + (r^2)))^2)), {r, 0.1, 0.8, 
            0.1}]], {t, -0.5, 0.5}];

Exemple 6

f[n_, t_] := 
      (((((1/2) Sin[(n + 1) π t] + Sin[π t]))^2)/(n + 1))

Plot[Evaluate[Table[f[n, t], {n, 1, 3}]], {t, -0.6, 0.6}];

Plot[Evaluate[Table[f[n, t], {n, 1, 10}]], {t, -0.6, 0.6}];

Exemple 7

f[n_, t_Integer] := n + 1;

f[n_, t_] := 
      ((((Sin[(n + 1) π t]/Sin[π t]))^2)/(n + 1))

Plot[Evaluate[Table[f[n, t], {n, 1, 3}]], {t, -0.6, 0.6}];

Clear[f]

f[n_][t_Integer] := 2 n + 1;

f[n_][t_] := Sin[(2 n + 1) π t]/Sin[π t]

Exemple 8

Plot[Evaluate[Table[f[n][t], {n, 1, 4}]], {t, -0.6, 0.6}];

Clear[f]

Exemple 9

f[n_][t_Integer] := 2 n + 1;

f[n_][t_] := 
      Max[Sin[(2 n + 1) π t]/Sin[π t], 
        Sin[(2 n) π t]/Sin[π t]]

Plot[Evaluate[Table[f[n][t], {n, 1, 3}]], {t, -0.6, 0.6}];

Exemple 10

f[n_][t_] := 
      Max[Sin[(2 n + 1) π t]/Sin[π t], 
        Sin[(2 n) π t]/Sin[π t], 0]

Plot[Evaluate[Table[f[n][t], {n, 1, 3}]], {t, -0.6, 0.6}];

f[n_][t_Integer] := 2 n + 1;

f [n_][t_] := Sin [(2 n + 1) Pi t / Sin [Pi t]

Plot[Evaluate[Table[f[n][t], {n, 1, 4}]], {t, -0.6, 0.6}, 
      PlotPoints -> 100];

Exemple 11

f[n_][t_Integer] := 2 n + 1;

f[n_][t_] := Sin[((2*n + 1)*Pi*t)/Sin[Pi*t]]

Plot[Evaluate[Table[f[n][t], {n, 1, 4}]], {t, -0.6, 0.6}, 
      PlotPoints -> 100];

Sin[(2 n + 1) π t]/Sin[π t] 
      de :   Sin[(((2 n + 1) π t)/Sin)[π t]]

Clear[f]

Exemple 12

S[n_] := NSum[(1/(k (k + 3))), {k, 1, n}]

Table[S[n], {n, 1, 30}]

ListPlot[%];

Table[S[n], {n, 31, 50}]

ListPlot[{0.25, 0.35, 0.4055555555555555555, 0.4412698412698412698, 
        0.4662698412698412698, 0.4847883597883597884, 0.4990740740740740741, 
        0.5104377104377104376, 0.5196969696969696969, 0.5273892773892773892, 
        0.5338827838827838827, 0.5394383394383394383, 0.544246031746031746, 
        0.5484477124183006536, 0.5521514161220043573, 0.5554408898062148836, 
        0.5583820662768031189, 0.5610275689223057644, 0.5634199134199134199, 
        0.5655938264633916807, 0.5675779534475186649, 0.5693961352657004831, 
        0.5710683760683760684, 0.5726115859449192782, 0.5740401573256721763, 
        0.5753664172720084004, 0.5766009851727856728, 0.5777530589051600427, 
        0.5788306451117944868, 0.5798407461216942662, 0.5807895127249540132, 
        0.5816823698676895435, 0.5825241207093447525, 0.5833190332688874588, 
        0.5840709129680752016, 0.5847831636802774767, 0.5854588393559140634, 
        0.5861006878796307352, 0.5867111884901054626, 0.5872925838389214457, 
        0.5878469075639593879, 0.5883760080930454385, 0.5888815692659353215, 
        0.5893651282601225631, 0.5898280912230771012, 0.590271746946228841, 
        0.5906972788611164705, 0.5911057755931375396, 0.5914982402713121882, 
        0.5918755987618745449}];

Exemple 13

Needs["Algebra`SymbolicSum`"]

(∑_(k = 1)^∞) (1/(k (k + 1)))

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

Exemple 14

(∑_(n = 1)^∞) (1/((2 n - 1) (2 n + 1)))

Exemple 15

(∑_(n = 1)^∞) (1/((3 n - 2) (3 n + 1)))

Exemple 16

(∑_(k = 1)^∞) (1/(k (k + 3)))

Exemple 17

(∑_(k = 1)^∞) (1/((2 k - 1) (2 k + 5)))

Exemple 18

(∑_(n = 1)^∞) (1/(n (n + 1) (n + 2)))

Exemple 19

(∑_(n = 1)^∞)(( (3^n) + (2^n))/(6^n))

Exemple 20

(∑_(n = 1)^∞)(( 2 n + 1)/((n^2) 
          (((n + 1))^2)))