---

Sea la sucesión (𝑎ₙ) definida por y 𝑎ₙ₊₁ = 6 − (8:𝑎ₙ)

a) Es constante para valores comprendidos entre (-∞,1]

b) Si 𝑎₁ = 3 la sucesión está acotada superiormente

c) Ninguna de las otras dos

Se puede estudiar el comportamiento ejecutando este trozo de código:

with Ada.Text_IO; use Ada.Text_IO;
with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
with Ada.Float_Text_IO; use Ada.Float_Text_IO;

procedure test is
    X: Float := 0.0;
begin
    for n in -100..10 loop
        x := 6.0-8.0/float(n);
        Put(n);
        Put(x);
        new_line;
        end loop;
end test;

       -100 6.08000E+00
        -99 6.08081E+00
        -98 6.08163E+00
        -97 6.08247E+00
        -96 6.08333E+00
        -95 6.08421E+00
        -94 6.08511E+00
        -93 6.08602E+00
        -92 6.08696E+00
        -91 6.08791E+00
        -90 6.08889E+00
        -89 6.08989E+00
        -88 6.09091E+00
        -87 6.09195E+00
        -86 6.09302E+00
        -85 6.09412E+00
        -84 6.09524E+00
        -83 6.09639E+00
        -82 6.09756E+00
        -81 6.09877E+00
        -80 6.10000E+00
        -79 6.10127E+00
        -78 6.10256E+00
        -77 6.10390E+00
        -76 6.10526E+00
        -75 6.10667E+00
        -74 6.10811E+00
        -73 6.10959E+00
        -72 6.11111E+00
        -71 6.11268E+00
        -70 6.11429E+00
        -69 6.11594E+00
        -68 6.11765E+00
        -67 6.11940E+00
        -66 6.12121E+00
        -65 6.12308E+00
        -64 6.12500E+00
        -63 6.12698E+00
        -62 6.12903E+00
        -61 6.13115E+00
        -60 6.13333E+00
        -59 6.13559E+00
        -58 6.13793E+00
        -57 6.14035E+00
        -56 6.14286E+00
        -55 6.14545E+00
        -54 6.14815E+00
        -53 6.15094E+00
        -52 6.15385E+00
        -51 6.15686E+00
        -50 6.16000E+00
        -49 6.16327E+00
        -48 6.16667E+00
        -47 6.17021E+00
        -46 6.17391E+00
        -45 6.17778E+00
        -44 6.18182E+00
        -43 6.18605E+00
        -42 6.19048E+00
        -41 6.19512E+00
        -40 6.20000E+00
        -39 6.20513E+00
        -38 6.21053E+00
        -37 6.21622E+00
        -36 6.22222E+00
        -35 6.22857E+00
        -34 6.23529E+00
        -33 6.24242E+00
        -32 6.25000E+00
        -31 6.25806E+00
        -30 6.26667E+00
        -29 6.27586E+00
        -28 6.28571E+00
        -27 6.29630E+00
        -26 6.30769E+00
        -25 6.32000E+00
        -24 6.33333E+00
        -23 6.34783E+00
        -22 6.36364E+00
        -21 6.38095E+00
        -20 6.40000E+00
        -19 6.42105E+00
        -18 6.44444E+00
        -17 6.47059E+00
        -16 6.50000E+00
        -15 6.53333E+00
        -14 6.57143E+00
        -13 6.61538E+00
        -12 6.66667E+00
        -11 6.72727E+00
        -10 6.80000E+00
         -9 6.88889E+00
         -8 7.00000E+00
         -7 7.14286E+00
         -6 7.33333E+00
         -5 7.60000E+00
         -4 8.00000E+00
         -3 8.66667E+00
         -2 1.00000E+01
         -1 1.40000E+01
          0-Inf********
          1-2.00000E+00
          2 2.00000E+00
          3 3.33333E+00
          4 4.00000E+00
          5 4.40000E+00
          6 4.66667E+00
          7 4.85714E+00
          8 5.00000E+00
          9 5.11111E+00
         10 5.20000E+00

Vemos que el comportamiento de la serie entre -100 y 1 no es constant para todo el intervalo.

Por ejemplo, para 𝑎₁ = -4, 𝑎₂ = 8, y 𝑎₃ = 5

with Ada.Text_IO; use Ada.Text_IO;
with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
with Ada.Float_Text_IO; use Ada.Float_Text_IO;

procedure test is
    X: Float := 3.0;
begin
    for n in 1..100 loop
        x := 6.0-8.0/x;
        Put(n);
        Put(x);
        new_line;
        end loop;
end test;

          1 3.33333E+00
          2 3.60000E+00
          3 3.77778E+00
          4 3.88235E+00
          5 3.93939E+00
          6 3.96923E+00
          7 3.98450E+00
          8 3.99222E+00
          9 3.99610E+00
         10 3.99805E+00
         11 3.99902E+00
         12 3.99951E+00
         13 3.99976E+00
         14 3.99988E+00
         15 3.99994E+00
         16 3.99997E+00
         17 3.99998E+00
         18 3.99999E+00
         19 4.00000E+00
         20 4.00000E+00

Vemos que el comportamiento de la sucesión tiende a 4 con 𝑎₁=3. Por tanto está acotada superiormente.

b) ✔

---

$\lim_{n\rightarrow ∞}\frac{\sum_{k=1} k^n }{n^p}$

---

Dada la sucesión $a_n = (1+n̈)+(-1)ⁿ(1-3n̈)$

$a_n = (1+n̈)+(-1)ⁿ(1-3n̈)$

Pares/Impares
$a_p = (1+p̈)+(1-3p̈)$
$a_m = (1+m̈)-(1-3m̈)$

$a_p∞ = (1)+(1) = 2$
$a_m∞ = (1)-(1)= 0$

---

$\lim_{n\rightarrow ∞}\frac{n! }{n^n}$

$\frac{n! }{n^n}$
$\frac{n·(n-1)...3·2·1 }{n·n·n·n·n...n}$
$\frac{n}{n} ·...· \frac{3}{n} \frac{2}{n} \frac{1}{n}$
$a_i = i/n$
$a_i ≤ 1$
$∏a_i ⟶ 0$