The series $frac{1}{2}+frac{2}{5}+frac{3}{11}+frac{4}{23}+…$
$begingroup$
Consider the expression $frac{1}{2}+frac{2}{5}+frac{3}{11}+frac{4}{23}+...$
Denote the numerator and the denominator of the $j^text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, $D_1=2$, and, for every $j>1$, $$
N_j=
N_{j-1}+1qquad D_j=
2D_{j-1}+1$$
What is the $50^text{th}$ term?
Must we evaluate that term-by-term until we reach the $50^text{th}$ term?
What is the sum of the first $25$ term?
Must we add them one-by-one?
What is the exact value of the sum to $infty$?
sequences-and-series convergence summation recurrence-relations
$endgroup$
|
show 4 more comments
$begingroup$
Consider the expression $frac{1}{2}+frac{2}{5}+frac{3}{11}+frac{4}{23}+...$
Denote the numerator and the denominator of the $j^text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, $D_1=2$, and, for every $j>1$, $$
N_j=
N_{j-1}+1qquad D_j=
2D_{j-1}+1$$
What is the $50^text{th}$ term?
Must we evaluate that term-by-term until we reach the $50^text{th}$ term?
What is the sum of the first $25$ term?
Must we add them one-by-one?
What is the exact value of the sum to $infty$?
sequences-and-series convergence summation recurrence-relations
$endgroup$
2
$begingroup$
Answers to your question: no and no.
$endgroup$
– Yves Daoust
Dec 9 '18 at 9:19
1
$begingroup$
Did you mean to write $D_j = 2D_{j-1}+1$ for $j > 1$?
$endgroup$
– JimmyK4542
Dec 9 '18 at 9:20
$begingroup$
@JimmyK4542 Yes, I did.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:23
$begingroup$
@AweKumarJha , $frac{50}{523}$ is not true, the $50^text{th}$ term is much less than that.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:26
$begingroup$
Then it just looks like a very similar series involving primes. Try using the method of generating functions to convert the recursion into explicit formula.
$endgroup$
– Awe Kumar Jha
Dec 9 '18 at 9:28
|
show 4 more comments
$begingroup$
Consider the expression $frac{1}{2}+frac{2}{5}+frac{3}{11}+frac{4}{23}+...$
Denote the numerator and the denominator of the $j^text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, $D_1=2$, and, for every $j>1$, $$
N_j=
N_{j-1}+1qquad D_j=
2D_{j-1}+1$$
What is the $50^text{th}$ term?
Must we evaluate that term-by-term until we reach the $50^text{th}$ term?
What is the sum of the first $25$ term?
Must we add them one-by-one?
What is the exact value of the sum to $infty$?
sequences-and-series convergence summation recurrence-relations
$endgroup$
Consider the expression $frac{1}{2}+frac{2}{5}+frac{3}{11}+frac{4}{23}+...$
Denote the numerator and the denominator of the $j^text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, $D_1=2$, and, for every $j>1$, $$
N_j=
N_{j-1}+1qquad D_j=
2D_{j-1}+1$$
What is the $50^text{th}$ term?
Must we evaluate that term-by-term until we reach the $50^text{th}$ term?
What is the sum of the first $25$ term?
Must we add them one-by-one?
What is the exact value of the sum to $infty$?
sequences-and-series convergence summation recurrence-relations
sequences-and-series convergence summation recurrence-relations
edited Dec 9 '18 at 9:56
Did
247k23222458
247k23222458
asked Dec 9 '18 at 9:14
Hussain-AlqatariHussain-Alqatari
3187
3187
2
$begingroup$
Answers to your question: no and no.
$endgroup$
– Yves Daoust
Dec 9 '18 at 9:19
1
$begingroup$
Did you mean to write $D_j = 2D_{j-1}+1$ for $j > 1$?
$endgroup$
– JimmyK4542
Dec 9 '18 at 9:20
$begingroup$
@JimmyK4542 Yes, I did.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:23
$begingroup$
@AweKumarJha , $frac{50}{523}$ is not true, the $50^text{th}$ term is much less than that.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:26
$begingroup$
Then it just looks like a very similar series involving primes. Try using the method of generating functions to convert the recursion into explicit formula.
$endgroup$
– Awe Kumar Jha
Dec 9 '18 at 9:28
|
show 4 more comments
2
$begingroup$
Answers to your question: no and no.
$endgroup$
– Yves Daoust
Dec 9 '18 at 9:19
1
$begingroup$
Did you mean to write $D_j = 2D_{j-1}+1$ for $j > 1$?
$endgroup$
– JimmyK4542
Dec 9 '18 at 9:20
$begingroup$
@JimmyK4542 Yes, I did.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:23
$begingroup$
@AweKumarJha , $frac{50}{523}$ is not true, the $50^text{th}$ term is much less than that.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:26
$begingroup$
Then it just looks like a very similar series involving primes. Try using the method of generating functions to convert the recursion into explicit formula.
$endgroup$
– Awe Kumar Jha
Dec 9 '18 at 9:28
2
2
$begingroup$
Answers to your question: no and no.
$endgroup$
– Yves Daoust
Dec 9 '18 at 9:19
$begingroup$
Answers to your question: no and no.
$endgroup$
– Yves Daoust
Dec 9 '18 at 9:19
1
1
$begingroup$
Did you mean to write $D_j = 2D_{j-1}+1$ for $j > 1$?
$endgroup$
– JimmyK4542
Dec 9 '18 at 9:20
$begingroup$
Did you mean to write $D_j = 2D_{j-1}+1$ for $j > 1$?
$endgroup$
– JimmyK4542
Dec 9 '18 at 9:20
$begingroup$
@JimmyK4542 Yes, I did.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:23
$begingroup$
@JimmyK4542 Yes, I did.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:23
$begingroup$
@AweKumarJha , $frac{50}{523}$ is not true, the $50^text{th}$ term is much less than that.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:26
$begingroup$
@AweKumarJha , $frac{50}{523}$ is not true, the $50^text{th}$ term is much less than that.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:26
$begingroup$
Then it just looks like a very similar series involving primes. Try using the method of generating functions to convert the recursion into explicit formula.
$endgroup$
– Awe Kumar Jha
Dec 9 '18 at 9:28
$begingroup$
Then it just looks like a very similar series involving primes. Try using the method of generating functions to convert the recursion into explicit formula.
$endgroup$
– Awe Kumar Jha
Dec 9 '18 at 9:28
|
show 4 more comments
3 Answers
3
active
oldest
votes
$begingroup$
The $n^{th}$ term is $frac{n}{3cdot 2^{n-1}-1}$, as can easily be proven by induction. [You can guess this by simply looking at the first two terms as you know it has to be of the form $ccdot2^n+d$]
$therefore 50^{th}$ term$=frac{50}{3cdot 2^{49}-1}$.
I tried all the methods I know(which includes generating functions, bruteforce calculation, CAS). The sum does not have a closed form formula.(As is the case with most nontrivial rapidly converging sums). You may however calculate the sum with arbitrary precision pretty easily.
$endgroup$
$begingroup$
This is helpful. Thank you. What about my second and the third questions?
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:30
$begingroup$
I am trying those parts. Will edit as soon as I get something useful. As of now, I think there is no closed form for the summation.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:32
1
$begingroup$
Please usecdot
instead of.
. The notation is really confusing.
$endgroup$
– Kemono Chen
Dec 9 '18 at 9:35
$begingroup$
@KemonoChen, fixed it.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:36
1
$begingroup$
Appreciate the work, THANKS!
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 10:36
|
show 1 more comment
$begingroup$
If $d_{j+1}=2d_j+1$ then $d_{j+1}+1=2(d_j+1)Rightarrow d_n+1=(d_1+1)2^{n-1}=3cdot2^{n-1}$.
So 50th term is $frac{50}{3cdot2^{49}-1}$
$endgroup$
add a comment |
$begingroup$
$D_j=2D_{j-1}+1\ =2(2D_{j-2}+1)+1\ =4D_{j-2}+1+2\ =4(2D_{j-3}+1)+1+2\ vdots\ =2^kD_{j-k}+2^k-1\ =2^{j-1}D_1+2^{j-1}-1\ =3cdot2^{j-1}-1$
$s_n=N_n/D_n=displaystylefrac n{3cdot2^{n-1}-1}, nge1$
$s_n<displaystylefrac n{3cdot2^{n-1}-2^{n-1}}=frac n{2^n}$
$displaystylesum_1^infty s_n<sum_1^infty frac n{2^n}$ which is an AP-GP series
$sum_1^infty frac n{2^n}=frac12+frac24+frac38...$
$frac12sum_1^infty frac n{2^n}=0+frac14+frac28+frac3{16}...$
$sum_1^infty frac n{2^n}-frac12sum_1^infty frac n{2^n}=frac12sum_1^infty frac n{2^n}=frac12+frac14+frac18...=1$
$displaystyleimplies0<sum_1^infty s_n<2$
$endgroup$
$begingroup$
The sum is not less than $4/3$, it is $1.5997809dots$.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:58
$begingroup$
My bad, I'll recheck what I wrote.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:00
$begingroup$
Actually, s_n > n/(3⋅2^n−1)
$endgroup$
– Ankit Kumar
Dec 9 '18 at 10:01
$begingroup$
Yeah, thanks for pointing it out.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:03
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The $n^{th}$ term is $frac{n}{3cdot 2^{n-1}-1}$, as can easily be proven by induction. [You can guess this by simply looking at the first two terms as you know it has to be of the form $ccdot2^n+d$]
$therefore 50^{th}$ term$=frac{50}{3cdot 2^{49}-1}$.
I tried all the methods I know(which includes generating functions, bruteforce calculation, CAS). The sum does not have a closed form formula.(As is the case with most nontrivial rapidly converging sums). You may however calculate the sum with arbitrary precision pretty easily.
$endgroup$
$begingroup$
This is helpful. Thank you. What about my second and the third questions?
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:30
$begingroup$
I am trying those parts. Will edit as soon as I get something useful. As of now, I think there is no closed form for the summation.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:32
1
$begingroup$
Please usecdot
instead of.
. The notation is really confusing.
$endgroup$
– Kemono Chen
Dec 9 '18 at 9:35
$begingroup$
@KemonoChen, fixed it.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:36
1
$begingroup$
Appreciate the work, THANKS!
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 10:36
|
show 1 more comment
$begingroup$
The $n^{th}$ term is $frac{n}{3cdot 2^{n-1}-1}$, as can easily be proven by induction. [You can guess this by simply looking at the first two terms as you know it has to be of the form $ccdot2^n+d$]
$therefore 50^{th}$ term$=frac{50}{3cdot 2^{49}-1}$.
I tried all the methods I know(which includes generating functions, bruteforce calculation, CAS). The sum does not have a closed form formula.(As is the case with most nontrivial rapidly converging sums). You may however calculate the sum with arbitrary precision pretty easily.
$endgroup$
$begingroup$
This is helpful. Thank you. What about my second and the third questions?
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:30
$begingroup$
I am trying those parts. Will edit as soon as I get something useful. As of now, I think there is no closed form for the summation.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:32
1
$begingroup$
Please usecdot
instead of.
. The notation is really confusing.
$endgroup$
– Kemono Chen
Dec 9 '18 at 9:35
$begingroup$
@KemonoChen, fixed it.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:36
1
$begingroup$
Appreciate the work, THANKS!
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 10:36
|
show 1 more comment
$begingroup$
The $n^{th}$ term is $frac{n}{3cdot 2^{n-1}-1}$, as can easily be proven by induction. [You can guess this by simply looking at the first two terms as you know it has to be of the form $ccdot2^n+d$]
$therefore 50^{th}$ term$=frac{50}{3cdot 2^{49}-1}$.
I tried all the methods I know(which includes generating functions, bruteforce calculation, CAS). The sum does not have a closed form formula.(As is the case with most nontrivial rapidly converging sums). You may however calculate the sum with arbitrary precision pretty easily.
$endgroup$
The $n^{th}$ term is $frac{n}{3cdot 2^{n-1}-1}$, as can easily be proven by induction. [You can guess this by simply looking at the first two terms as you know it has to be of the form $ccdot2^n+d$]
$therefore 50^{th}$ term$=frac{50}{3cdot 2^{49}-1}$.
I tried all the methods I know(which includes generating functions, bruteforce calculation, CAS). The sum does not have a closed form formula.(As is the case with most nontrivial rapidly converging sums). You may however calculate the sum with arbitrary precision pretty easily.
edited Dec 9 '18 at 10:22
answered Dec 9 '18 at 9:28
Anubhab GhosalAnubhab Ghosal
82518
82518
$begingroup$
This is helpful. Thank you. What about my second and the third questions?
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:30
$begingroup$
I am trying those parts. Will edit as soon as I get something useful. As of now, I think there is no closed form for the summation.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:32
1
$begingroup$
Please usecdot
instead of.
. The notation is really confusing.
$endgroup$
– Kemono Chen
Dec 9 '18 at 9:35
$begingroup$
@KemonoChen, fixed it.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:36
1
$begingroup$
Appreciate the work, THANKS!
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 10:36
|
show 1 more comment
$begingroup$
This is helpful. Thank you. What about my second and the third questions?
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:30
$begingroup$
I am trying those parts. Will edit as soon as I get something useful. As of now, I think there is no closed form for the summation.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:32
1
$begingroup$
Please usecdot
instead of.
. The notation is really confusing.
$endgroup$
– Kemono Chen
Dec 9 '18 at 9:35
$begingroup$
@KemonoChen, fixed it.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:36
1
$begingroup$
Appreciate the work, THANKS!
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 10:36
$begingroup$
This is helpful. Thank you. What about my second and the third questions?
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:30
$begingroup$
This is helpful. Thank you. What about my second and the third questions?
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:30
$begingroup$
I am trying those parts. Will edit as soon as I get something useful. As of now, I think there is no closed form for the summation.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:32
$begingroup$
I am trying those parts. Will edit as soon as I get something useful. As of now, I think there is no closed form for the summation.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:32
1
1
$begingroup$
Please use
cdot
instead of .
. The notation is really confusing.$endgroup$
– Kemono Chen
Dec 9 '18 at 9:35
$begingroup$
Please use
cdot
instead of .
. The notation is really confusing.$endgroup$
– Kemono Chen
Dec 9 '18 at 9:35
$begingroup$
@KemonoChen, fixed it.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:36
$begingroup$
@KemonoChen, fixed it.
$endgroup$
– Anubhab Ghosal
Dec 9 '18 at 9:36
1
1
$begingroup$
Appreciate the work, THANKS!
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 10:36
$begingroup$
Appreciate the work, THANKS!
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 10:36
|
show 1 more comment
$begingroup$
If $d_{j+1}=2d_j+1$ then $d_{j+1}+1=2(d_j+1)Rightarrow d_n+1=(d_1+1)2^{n-1}=3cdot2^{n-1}$.
So 50th term is $frac{50}{3cdot2^{49}-1}$
$endgroup$
add a comment |
$begingroup$
If $d_{j+1}=2d_j+1$ then $d_{j+1}+1=2(d_j+1)Rightarrow d_n+1=(d_1+1)2^{n-1}=3cdot2^{n-1}$.
So 50th term is $frac{50}{3cdot2^{49}-1}$
$endgroup$
add a comment |
$begingroup$
If $d_{j+1}=2d_j+1$ then $d_{j+1}+1=2(d_j+1)Rightarrow d_n+1=(d_1+1)2^{n-1}=3cdot2^{n-1}$.
So 50th term is $frac{50}{3cdot2^{49}-1}$
$endgroup$
If $d_{j+1}=2d_j+1$ then $d_{j+1}+1=2(d_j+1)Rightarrow d_n+1=(d_1+1)2^{n-1}=3cdot2^{n-1}$.
So 50th term is $frac{50}{3cdot2^{49}-1}$
answered Dec 9 '18 at 9:39
MinzMinz
951127
951127
add a comment |
add a comment |
$begingroup$
$D_j=2D_{j-1}+1\ =2(2D_{j-2}+1)+1\ =4D_{j-2}+1+2\ =4(2D_{j-3}+1)+1+2\ vdots\ =2^kD_{j-k}+2^k-1\ =2^{j-1}D_1+2^{j-1}-1\ =3cdot2^{j-1}-1$
$s_n=N_n/D_n=displaystylefrac n{3cdot2^{n-1}-1}, nge1$
$s_n<displaystylefrac n{3cdot2^{n-1}-2^{n-1}}=frac n{2^n}$
$displaystylesum_1^infty s_n<sum_1^infty frac n{2^n}$ which is an AP-GP series
$sum_1^infty frac n{2^n}=frac12+frac24+frac38...$
$frac12sum_1^infty frac n{2^n}=0+frac14+frac28+frac3{16}...$
$sum_1^infty frac n{2^n}-frac12sum_1^infty frac n{2^n}=frac12sum_1^infty frac n{2^n}=frac12+frac14+frac18...=1$
$displaystyleimplies0<sum_1^infty s_n<2$
$endgroup$
$begingroup$
The sum is not less than $4/3$, it is $1.5997809dots$.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:58
$begingroup$
My bad, I'll recheck what I wrote.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:00
$begingroup$
Actually, s_n > n/(3⋅2^n−1)
$endgroup$
– Ankit Kumar
Dec 9 '18 at 10:01
$begingroup$
Yeah, thanks for pointing it out.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:03
add a comment |
$begingroup$
$D_j=2D_{j-1}+1\ =2(2D_{j-2}+1)+1\ =4D_{j-2}+1+2\ =4(2D_{j-3}+1)+1+2\ vdots\ =2^kD_{j-k}+2^k-1\ =2^{j-1}D_1+2^{j-1}-1\ =3cdot2^{j-1}-1$
$s_n=N_n/D_n=displaystylefrac n{3cdot2^{n-1}-1}, nge1$
$s_n<displaystylefrac n{3cdot2^{n-1}-2^{n-1}}=frac n{2^n}$
$displaystylesum_1^infty s_n<sum_1^infty frac n{2^n}$ which is an AP-GP series
$sum_1^infty frac n{2^n}=frac12+frac24+frac38...$
$frac12sum_1^infty frac n{2^n}=0+frac14+frac28+frac3{16}...$
$sum_1^infty frac n{2^n}-frac12sum_1^infty frac n{2^n}=frac12sum_1^infty frac n{2^n}=frac12+frac14+frac18...=1$
$displaystyleimplies0<sum_1^infty s_n<2$
$endgroup$
$begingroup$
The sum is not less than $4/3$, it is $1.5997809dots$.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:58
$begingroup$
My bad, I'll recheck what I wrote.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:00
$begingroup$
Actually, s_n > n/(3⋅2^n−1)
$endgroup$
– Ankit Kumar
Dec 9 '18 at 10:01
$begingroup$
Yeah, thanks for pointing it out.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:03
add a comment |
$begingroup$
$D_j=2D_{j-1}+1\ =2(2D_{j-2}+1)+1\ =4D_{j-2}+1+2\ =4(2D_{j-3}+1)+1+2\ vdots\ =2^kD_{j-k}+2^k-1\ =2^{j-1}D_1+2^{j-1}-1\ =3cdot2^{j-1}-1$
$s_n=N_n/D_n=displaystylefrac n{3cdot2^{n-1}-1}, nge1$
$s_n<displaystylefrac n{3cdot2^{n-1}-2^{n-1}}=frac n{2^n}$
$displaystylesum_1^infty s_n<sum_1^infty frac n{2^n}$ which is an AP-GP series
$sum_1^infty frac n{2^n}=frac12+frac24+frac38...$
$frac12sum_1^infty frac n{2^n}=0+frac14+frac28+frac3{16}...$
$sum_1^infty frac n{2^n}-frac12sum_1^infty frac n{2^n}=frac12sum_1^infty frac n{2^n}=frac12+frac14+frac18...=1$
$displaystyleimplies0<sum_1^infty s_n<2$
$endgroup$
$D_j=2D_{j-1}+1\ =2(2D_{j-2}+1)+1\ =4D_{j-2}+1+2\ =4(2D_{j-3}+1)+1+2\ vdots\ =2^kD_{j-k}+2^k-1\ =2^{j-1}D_1+2^{j-1}-1\ =3cdot2^{j-1}-1$
$s_n=N_n/D_n=displaystylefrac n{3cdot2^{n-1}-1}, nge1$
$s_n<displaystylefrac n{3cdot2^{n-1}-2^{n-1}}=frac n{2^n}$
$displaystylesum_1^infty s_n<sum_1^infty frac n{2^n}$ which is an AP-GP series
$sum_1^infty frac n{2^n}=frac12+frac24+frac38...$
$frac12sum_1^infty frac n{2^n}=0+frac14+frac28+frac3{16}...$
$sum_1^infty frac n{2^n}-frac12sum_1^infty frac n{2^n}=frac12sum_1^infty frac n{2^n}=frac12+frac14+frac18...=1$
$displaystyleimplies0<sum_1^infty s_n<2$
edited Dec 9 '18 at 10:18
answered Dec 9 '18 at 9:52
Shubham JohriShubham Johri
4,885717
4,885717
$begingroup$
The sum is not less than $4/3$, it is $1.5997809dots$.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:58
$begingroup$
My bad, I'll recheck what I wrote.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:00
$begingroup$
Actually, s_n > n/(3⋅2^n−1)
$endgroup$
– Ankit Kumar
Dec 9 '18 at 10:01
$begingroup$
Yeah, thanks for pointing it out.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:03
add a comment |
$begingroup$
The sum is not less than $4/3$, it is $1.5997809dots$.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:58
$begingroup$
My bad, I'll recheck what I wrote.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:00
$begingroup$
Actually, s_n > n/(3⋅2^n−1)
$endgroup$
– Ankit Kumar
Dec 9 '18 at 10:01
$begingroup$
Yeah, thanks for pointing it out.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:03
$begingroup$
The sum is not less than $4/3$, it is $1.5997809dots$.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:58
$begingroup$
The sum is not less than $4/3$, it is $1.5997809dots$.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:58
$begingroup$
My bad, I'll recheck what I wrote.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:00
$begingroup$
My bad, I'll recheck what I wrote.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:00
$begingroup$
Actually, s_n > n/(3⋅2^n−1)
$endgroup$
– Ankit Kumar
Dec 9 '18 at 10:01
$begingroup$
Actually, s_n > n/(3⋅2^n−1)
$endgroup$
– Ankit Kumar
Dec 9 '18 at 10:01
$begingroup$
Yeah, thanks for pointing it out.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:03
$begingroup$
Yeah, thanks for pointing it out.
$endgroup$
– Shubham Johri
Dec 9 '18 at 10:03
add a comment |
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$begingroup$
Answers to your question: no and no.
$endgroup$
– Yves Daoust
Dec 9 '18 at 9:19
1
$begingroup$
Did you mean to write $D_j = 2D_{j-1}+1$ for $j > 1$?
$endgroup$
– JimmyK4542
Dec 9 '18 at 9:20
$begingroup$
@JimmyK4542 Yes, I did.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:23
$begingroup$
@AweKumarJha , $frac{50}{523}$ is not true, the $50^text{th}$ term is much less than that.
$endgroup$
– Hussain-Alqatari
Dec 9 '18 at 9:26
$begingroup$
Then it just looks like a very similar series involving primes. Try using the method of generating functions to convert the recursion into explicit formula.
$endgroup$
– Awe Kumar Jha
Dec 9 '18 at 9:28