$x_1=1,x_n=x_{n+1}+\ln (1+x_{n+1})$, prove
$x_n\leq\frac{1}{2^{n-2}}$.
I proved $0<x_{n+1}<x_n$ by contradiction, and I also get $x_n\geq\frac{1}{2^{n-1}}$ by induction. I tried $\ln(1+x)\geq x-\frac{1}{2}x^2$, but it did not work. Could anyone help me? Thanks!
I tried a new way
$\forall \epsilon>0, \ln(1+x)>(1-\epsilon)x$, then $x_n>(2-\epsilon)x_{n+1}$, thus $x_{n+1}<\frac{1}{(2-\epsilon)2^{n-2}}\to \frac{1}{2^{n-1}}$.
Is it right?
On
the material you edited in is not enough. We do get an intermediate detail, $$ 0 < x_{n+1} \; < \; \frac{1}{1 + \sqrt{1 - \frac{x_n}{2} \; }} $$ which does convince me that, for example, $$ 2^n x_n $$ is bounded and has a limit, let us call the limit $C.$ The bad news is that they want $C < 4$ and possibly a little more than that information. More work is required. If $2^n x_n$ decreases with $n$ that would be enough. If, instead, $2^n x_n$ increases with $n$ as well as the limit $C \leq 4,$ that would do it.
ADDED: I did an experiment just using the rule $$ x_{n+1} \; = \; \frac{1}{1 + \sqrt{1 - \frac{x_n}{2} \; }} $$ which suggests that the original sequence has $2^n x_n$ increasing with limit below $3$
Mon May 7 11:16:13 PDT 2018
2 x_n : 0.585786437626905 2^n x_n : 2.34314575050762
3 x_n : 0.3182071694925709 2^n x_n : 2.545657355940568
4 x_n : 0.1659919135906575 2^n x_n : 2.655870617450521
5 x_n : 0.08479343860285271 2^n x_n : 2.713390035291287
6 x_n : 0.04285587582459973 2^n x_n : 2.742776052774383
7 x_n : 0.02154397361204903 2^n x_n : 2.757628622342276
8 x_n : 0.01080115303273365 2^n x_n : 2.765095176379814
9 x_n : 0.005407887829059747 2^n x_n : 2.768838568478591
10 x_n : 0.002705774218059653 2^n x_n : 2.770712799293084
11 x_n : 0.001353344994698495 2^n x_n : 2.771650549142518
12 x_n : 0.0006767870075126321 2^n x_n : 2.772119582771741
13 x_n : 0.0003384221361418738 2^n x_n : 2.77235413927423
14 x_n : 0.0001692182267730049 2^n x_n : 2.772471427448913
15 x_n : 8.46109031377349e-05 2^n x_n : 2.772530074017297
16 x_n : 4.230589901614034e-05 2^n x_n : 2.772559397921774
17 x_n : 2.115306137107151e-05 2^n x_n : 2.772574060029085
18 x_n : 1.057655865143398e-05 2^n x_n : 2.77258139112151
19 x_n : 5.288286317210035e-06 2^n x_n : 2.772585056677415
20 x_n : 2.644144906480589e-06 2^n x_n : 2.77258688945779
21 x_n : 1.322072890209476e-06 2^n x_n : 2.772587805848584
22 x_n : 6.610365543470697e-07 2^n x_n : 2.772588264044132
23 x_n : 3.305183044841222e-07 2^n x_n : 2.772588493141944
24 x_n : 1.652591590697085e-07 2^n x_n : 2.772588607690859
25 x_n : 8.262958124176617e-08 2^n x_n : 2.772588664965319
26 x_n : 4.131479104761107e-08 2^n x_n : 2.772588693602549
27 x_n : 2.065739563048754e-08 2^n x_n : 2.772588707921165
28 x_n : 1.032869784191427e-08 2^n x_n : 2.772588715080472
29 x_n : 5.164348927624759e-09 2^n x_n : 2.772588718660126
30 x_n : 2.582174465479286e-09 2^n x_n : 2.772588720449953
31 x_n : 1.291087233156369e-09 2^n x_n : 2.772588721344867
32 x_n : 6.455436166823664e-10 2^n x_n : 2.772588721792324
33 x_n : 3.227718083672286e-10 2^n x_n : 2.772588722016052
34 x_n : 1.613859041901257e-10 2^n x_n : 2.772588722127916
35 x_n : 8.069295209669067e-11 2^n x_n : 2.772588722183849
36 x_n : 4.03464760487523e-11 2^n x_n : 2.772588722211815
37 x_n : 2.017323802447789e-11 2^n x_n : 2.772588722225798
38 x_n : 1.008661901226438e-11 2^n x_n : 2.772588722232789
39 x_n : 5.043309506138548e-12 2^n x_n : 2.772588722236285
40 x_n : 2.521654753070864e-12 2^n x_n : 2.772588722238033
41 x_n : 1.260827376535829e-12 2^n x_n : 2.772588722238907
42 x_n : 6.30413688268014e-13 2^n x_n : 2.772588722239344
43 x_n : 3.152068441340318e-13 2^n x_n : 2.772588722239562
44 x_n : 1.576034220670221e-13 2^n x_n : 2.772588722239671
45 x_n : 7.880171103351262e-14 2^n x_n : 2.772588722239726
46 x_n : 3.94008555167567e-14 2^n x_n : 2.772588722239754
47 x_n : 1.970042775837845e-14 2^n x_n : 2.772588722239767
48 x_n : 9.850213879189249e-15 2^n x_n : 2.772588722239774
49 x_n : 4.925106939594631e-15 2^n x_n : 2.772588722239778
50 x_n : 2.462553469797317e-15 2^n x_n : 2.77258872223978
51 x_n : 1.231276734898659e-15 2^n x_n : 2.772588722239781
52 x_n : 6.156383674493296e-16 2^n x_n : 2.772588722239781
53 x_n : 3.078191837246648e-16 2^n x_n : 2.772588722239782
54 x_n : 1.539095918623324e-16 2^n x_n : 2.772588722239782
55 x_n : 7.695479593116621e-17 2^n x_n : 2.772588722239782
56 x_n : 3.84773979655831e-17 2^n x_n : 2.772588722239782
57 x_n : 1.923869898279155e-17 2^n x_n : 2.772588722239782
58 x_n : 9.619349491395776e-18 2^n x_n : 2.772588722239782
59 x_n : 4.809674745697888e-18 2^n x_n : 2.772588722239782
60 x_n : 2.404837372848944e-18 2^n x_n : 2.772588722239782
61 x_n : 1.202418686424472e-18 2^n x_n : 2.772588722239782
62 x_n : 6.01209343212236e-19 2^n x_n : 2.772588722239782
63 x_n : 3.00604671606118e-19 2^n x_n : 2.772588722239782
64 x_n : 1.50302335803059e-19 2^n x_n : 2.772588722239782
65 x_n : 7.51511679015295e-20 2^n x_n : 2.772588722239782
66 x_n : 3.757558395076475e-20 2^n x_n : 2.772588722239782
67 x_n : 1.878779197538238e-20 2^n x_n : 2.772588722239782
68 x_n : 9.393895987691188e-21 2^n x_n : 2.772588722239782
69 x_n : 4.696947993845594e-21 2^n x_n : 2.772588722239782
70 x_n : 2.348473996922797e-21 2^n x_n : 2.772588722239782
71 x_n : 1.174236998461398e-21 2^n x_n : 2.772588722239782
72 x_n : 5.871184992306992e-22 2^n x_n : 2.772588722239782
73 x_n : 2.935592496153496e-22 2^n x_n : 2.772588722239782
74 x_n : 1.467796248076748e-22 2^n x_n : 2.772588722239782
75 x_n : 7.33898124038374e-23 2^n x_n : 2.772588722239782
76 x_n : 3.66949062019187e-23 2^n x_n : 2.772588722239782
77 x_n : 1.834745310095935e-23 2^n x_n : 2.772588722239782
78 x_n : 9.173726550479675e-24 2^n x_n : 2.772588722239782
79 x_n : 4.586863275239838e-24 2^n x_n : 2.772588722239782
80 x_n : 2.293431637619919e-24 2^n x_n : 2.772588722239782
81 x_n : 1.146715818809959e-24 2^n x_n : 2.772588722239782
82 x_n : 5.733579094049797e-25 2^n x_n : 2.772588722239782
83 x_n : 2.866789547024899e-25 2^n x_n : 2.772588722239782
84 x_n : 1.433394773512449e-25 2^n x_n : 2.772588722239782
85 x_n : 7.166973867562246e-26 2^n x_n : 2.772588722239782
86 x_n : 3.583486933781123e-26 2^n x_n : 2.772588722239782
87 x_n : 1.791743466890562e-26 2^n x_n : 2.772588722239782
88 x_n : 8.958717334452808e-27 2^n x_n : 2.772588722239782
89 x_n : 4.479358667226404e-27 2^n x_n : 2.772588722239782
90 x_n : 2.239679333613202e-27 2^n x_n : 2.772588722239782
91 x_n : 1.119839666806601e-27 2^n x_n : 2.772588722239782
92 x_n : 5.599198334033005e-28 2^n x_n : 2.772588722239782
93 x_n : 2.799599167016502e-28 2^n x_n : 2.772588722239782
94 x_n : 1.399799583508251e-28 2^n x_n : 2.772588722239782
95 x_n : 6.998997917541256e-29 2^n x_n : 2.772588722239782
96 x_n : 3.499498958770628e-29 2^n x_n : 2.772588722239782
97 x_n : 1.749749479385314e-29 2^n x_n : 2.772588722239782
98 x_n : 8.74874739692657e-30 2^n x_n : 2.772588722239782
99 x_n : 4.374373698463285e-30 2^n x_n : 2.772588722239782
100 x_n : 2.187186849231643e-30 2^n x_n : 2.772588722239782
Mon May 7 11:16:13 PDT 2018
Not a complete answer but an upper bound.
If you use the inequality: $\log(1+x)\geq\dfrac{x}{1+x}$ for $x>0,$
Then, you have $x_n\geq x_{n+1}+\dfrac{x_{n+1}}{1+x_{n+1}} = x_{n+1}\dfrac{2+x_{n+1}}{1+x_{n+1}}$ or $$x_{n+1}^2-x_{n+1}(x_n-2)-x_n\leq 0.$$ This implies $x_{n+1}\leq \dfrac{x_n-2+\sqrt{x_n^2+4}}{2}=\dfrac{2x_n}{\sqrt{x_n^2+4}-x_n+2}$. But I checked that if you used induction directly now, then it would not work. Maybe there is a way to use this bound differently and get your result.
well, I think I misunderstand this question, sorry.
as you said
$x_{n+1}\ge ln(1+x_{n+1})\geq x-\frac{1}{2}x^2$
we can get $x_n\in(\frac{1}{2^{n-1}}, 2-\sqrt{4-2x_{n-1}})$
Thus we need to show $2-\sqrt{4-2x_{n-1}}\le \frac{1}{2^{n-2}}$
in fact, the above $2-\sqrt{4-2x_{n-1}}$ is determined by $ 2-\sqrt{2\sqrt{2\sqrt{2...}}}$ which you can check with $\frac{1}{2^{n-2}} $