299. A class of the recurrent sequencies.
by Virgil Nicula, Jul 18, 2011, 2:14 AM
Theorem. Let 
be a continue function for which 
and exists 
so that 
.
Then the recurrent sequence
and 
for any 
has the properties 
and 
.
Proof. Let
be a function for which 
and 
for any 
. Observe that the function 
is continue and for any
we have 
( is a dynamic function). Thus, the sequence 
is defined correctly : and 
,
where . For any we have 
, what means the sequence 
(strict decreasing). For 
in the recurrent relation obtain that
, i.e. 
. Indeed, if 
, then 
, absurd. In conclusion, 
. Observe that
and 
. From the Stolz's theorem obtain that 
, i.e. 
.
Particular case. For the function
we have 
and 
, i.e. 
and 
. In conclusion, the sequence 
,
i.e.
has the properties and 
, i.e. 
, Vezi articolul meu din G.M.B. - O clasa de siruri recurente.
Examples. Ascertain the limits of the following strings 
, where 
:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
!2.
13.
14.
15.
16.
17.
18.
19.![$x_1>1,\ x_{n+1}=\sqrt [3] {3x_n-2},\ y_n=n(x_n-1)\ .$](//latex.artofproblemsolving.com/3/4/4/344a8ccb25f7ef8fe2046f2120a11c3c3e26974e.png)
20.
21.
.
See here and here
be a continue function for which 
and exists 
so that 
.Then the recurrent sequence
and 
for any 
has the properties 
and 
.Proof. Let
be a function for which 
and 
for any 
. Observe that the function 
is continue and for any we have 
( is a dynamic function). Thus, the sequence 
is defined correctly : and 
,where
. For any we have 
, what means the sequence 
(strict decreasing). For 
in the recurrent relation obtain that
, i.e. 
. Indeed, if 
, then 
, absurd. In conclusion, 
. Observe that
and 
. From the Stolz's theorem obtain that 
, i.e. 
.Particular case. For the function
we have 
and 
, i.e. 
and 
. In conclusion, the sequence 
,i.e.
has the properties and 
, i.e. 
, Vezi articolul meu din G.M.B. - O clasa de siruri recurente.
Examples. Ascertain the limits of the following strings 
, where 
:1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
!2.
13.
14.
15.
16.
17.
18.
19.
![$x_1>1,\ x_{n+1}=\sqrt [3] {3x_n-2},\ y_n=n(x_n-1)\ .$](http://latex.artofproblemsolving.com/3/4/4/344a8ccb25f7ef8fe2046f2120a11c3c3e26974e.png)
20.
21.
.See here and here
This post has been edited 21 times. Last edited by Virgil Nicula, Nov 21, 2015, 7:48 AM
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