Why the value of this kind of sequence always get bigger and then smaller
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I tried to use $1$, $2$ and $3$ into this formula, and for all the answer will get bigger first than smaller. I want to know the reason for it.
sequences-and-series
edited Sep 7 at 10:30
user220178
1768
asked Sep 7 at 9:40
teng
42
add a comment |Â
up vote
0
down vote
favorite
I tried to use $1$, $2$ and $3$ into this formula, and for all the answer will get bigger first than smaller. I want to know the reason for it.
sequences-and-series
edited Sep 7 at 10:30
user220178
1768
asked Sep 7 at 9:40
teng
42
The sequences are recursive sequences, as you might see. For example, in the first case, $$ x_n+1 = 1 + frac1x_n qquad textwith x_0 = 1 $$ So which one is bigger, $x_n$ or $x_n+1$? The limit of the sequence is $$ lim_x to infty x_n = frac1+sqrt52 = phi $$ and if $x_n < phi Rightarrow x_n+1 > phi$ etc ... You can see this by comparing the functions $$ f_1(x) = x $$ and $$ f_2(x) = 1+ frac1x $$ and figuring out when $f_1(x) > f_2(x)$.
– Matti P.
Sep 7 at 10:00
The sequence contains the convergents of the (simple) continued fraction of the golden ratio $ varphi $. Such a sequence always alternates between values above and values below the given number.
– Peter
Sep 7 at 11:08
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I tried to use $1$, $2$ and $3$ into this formula, and for all the answer will get bigger first than smaller. I want to know the reason for it.
sequences-and-series
edited Sep 7 at 10:30
user220178
1768
asked Sep 7 at 9:40
teng
42
I tried to use $1$, $2$ and $3$ into this formula, and for all the answer will get bigger first than smaller. I want to know the reason for it.
sequences-and-series
sequences-and-series
edited Sep 7 at 10:30
user220178
1768
asked Sep 7 at 9:40
teng
42
edited Sep 7 at 10:30
user220178
1768
asked Sep 7 at 9:40
teng
42
edited Sep 7 at 10:30
user220178
1768
edited Sep 7 at 10:30
user220178
1768
edited Sep 7 at 10:30
user220178
1768
1768
asked Sep 7 at 9:40
teng
42
asked Sep 7 at 9:40
teng
42
asked Sep 7 at 9:40
teng
42
42
The sequences are recursive sequences, as you might see. For example, in the first case, $$ x_n+1 = 1 + frac1x_n qquad textwith x_0 = 1 $$ So which one is bigger, $x_n$ or $x_n+1$? The limit of the sequence is $$ lim_x to infty x_n = frac1+sqrt52 = phi $$ and if $x_n < phi Rightarrow x_n+1 > phi$ etc ... You can see this by comparing the functions $$ f_1(x) = x $$ and $$ f_2(x) = 1+ frac1x $$ and figuring out when $f_1(x) > f_2(x)$.
– Matti P.
Sep 7 at 10:00
The sequence contains the convergents of the (simple) continued fraction of the golden ratio $ varphi $. Such a sequence always alternates between values above and values below the given number.
– Peter
Sep 7 at 11:08
add a comment |Â
The sequences are recursive sequences, as you might see. For example, in the first case, $$ x_n+1 = 1 + frac1x_n qquad textwith x_0 = 1 $$ So which one is bigger, $x_n$ or $x_n+1$? The limit of the sequence is $$ lim_x to infty x_n = frac1+sqrt52 = phi $$ and if $x_n < phi Rightarrow x_n+1 > phi$ etc ... You can see this by comparing the functions $$ f_1(x) = x $$ and $$ f_2(x) = 1+ frac1x $$ and figuring out when $f_1(x) > f_2(x)$.
– Matti P.
Sep 7 at 10:00
The sequence contains the convergents of the (simple) continued fraction of the golden ratio $ varphi $. Such a sequence always alternates between values above and values below the given number.
– Peter
Sep 7 at 11:08
The sequences are recursive sequences, as you might see. For example, in the first case, $$ x_n+1 = 1 + frac1x_n qquad textwith x_0 = 1 $$ So which one is bigger, $x_n$ or $x_n+1$? The limit of the sequence is $$ lim_x to infty x_n = frac1+sqrt52 = phi $$ and if $x_n < phi Rightarrow x_n+1 > phi$ etc ... You can see this by comparing the functions $$ f_1(x) = x $$ and $$ f_2(x) = 1+ frac1x $$ and figuring out when $f_1(x) > f_2(x)$.
– Matti P.
Sep 7 at 10:00
The sequences are recursive sequences, as you might see. For example, in the first case, $$ x_n+1 = 1 + frac1x_n qquad textwith x_0 = 1 $$ So which one is bigger, $x_n$ or $x_n+1$? The limit of the sequence is $$ lim_x to infty x_n = frac1+sqrt52 = phi $$ and if $x_n < phi Rightarrow x_n+1 > phi$ etc ... You can see this by comparing the functions $$ f_1(x) = x $$ and $$ f_2(x) = 1+ frac1x $$ and figuring out when $f_1(x) > f_2(x)$.
– Matti P.
Sep 7 at 10:00
The sequence contains the convergents of the (simple) continued fraction of the golden ratio $ varphi $. Such a sequence always alternates between values above and values below the given number.
– Peter
Sep 7 at 11:08
The sequence contains the convergents of the (simple) continued fraction of the golden ratio $ varphi $. Such a sequence always alternates between values above and values below the given number.
– Peter
Sep 7 at 11:08
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
These are simple continued fractions. This from wikipedia tells you the approximation errors alternate in sign:
Corollary 1: The even convergents continually increase, but are
always less than $x$.
Corollary 2: The odd convergents continually decrease, but are always
greater than $x$ .
(Good for you for noticing this.)
https://en.wikipedia.org/wiki/Continued_fraction#Properties
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
These are simple continued fractions. This from wikipedia tells you the approximation errors alternate in sign:
Corollary 1: The even convergents continually increase, but are
always less than $x$.
Corollary 2: The odd convergents continually decrease, but are always
greater than $x$ .
(Good for you for noticing this.)
https://en.wikipedia.org/wiki/Continued_fraction#Properties
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
add a comment |Â
up vote
0
down vote
These are simple continued fractions. This from wikipedia tells you the approximation errors alternate in sign:
Corollary 1: The even convergents continually increase, but are
always less than $x$.
Corollary 2: The odd convergents continually decrease, but are always
greater than $x$ .
(Good for you for noticing this.)
https://en.wikipedia.org/wiki/Continued_fraction#Properties
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
add a comment |Â
up vote
0
down vote
up vote
0
down vote
These are simple continued fractions. This from wikipedia tells you the approximation errors alternate in sign:
Corollary 1: The even convergents continually increase, but are
always less than $x$.
Corollary 2: The odd convergents continually decrease, but are always
greater than $x$ .
(Good for you for noticing this.)
https://en.wikipedia.org/wiki/Continued_fraction#Properties
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
These are simple continued fractions. This from wikipedia tells you the approximation errors alternate in sign:
Corollary 1: The even convergents continually increase, but are
always less than $x$.
Corollary 2: The odd convergents continually decrease, but are always
greater than $x$ .
(Good for you for noticing this.)
https://en.wikipedia.org/wiki/Continued_fraction#Properties
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
edited Sep 7 at 12:17
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
answered Sep 7 at 12:05
Ethan Bolker
36.7k54299
36.7k54299
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2908451%2fwhy-the-value-of-this-kind-of-sequence-always-get-bigger-and-then-smaller%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
The sequences are recursive sequences, as you might see. For example, in the first case, $$ x_n+1 = 1 + frac1x_n qquad textwith x_0 = 1 $$ So which one is bigger, $x_n$ or $x_n+1$? The limit of the sequence is $$ lim_x to infty x_n = frac1+sqrt52 = phi $$ and if $x_n < phi Rightarrow x_n+1 > phi$ etc ... You can see this by comparing the functions $$ f_1(x) = x $$ and $$ f_2(x) = 1+ frac1x $$ and figuring out when $f_1(x) > f_2(x)$.
– Matti P.
Sep 7 at 10:00
The sequence contains the convergents of the (simple) continued fraction of the golden ratio $ varphi $. Such a sequence always alternates between values above and values below the given number.
– Peter
Sep 7 at 11:08