Understanding Kronecker Delta Function for Faulhaber's Formula
$begingroup$
I am trying to use Faulhaber's formula to determine partial sums of a power series.
Faulhaber's formula is given by
$sum_{k=1}^{n}{k^{p}} = frac{1}{p+1}sum_{i=1}^{p+1}{(-1)^{delta_ip}{p+1choose i}}B_{p+1-i}n^{i}$ where $delta_ip$ is the Kronecker delta function and $B_{p+1-i}$ is the $p+1-i$th Bernoulli number.
My question is, what do I use for $i$ in the Kronecker delta function when using this formula?
For example, I am trying to derive the partial sum of the power series $sum_{k=1}^{n}{k^2}$ but I don't see what I would use for $i$.
calculus sequences-and-series summation binomial-coefficients
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
$endgroup$
add a comment |
$begingroup$
I am trying to use Faulhaber's formula to determine partial sums of a power series.
Faulhaber's formula is given by
$sum_{k=1}^{n}{k^{p}} = frac{1}{p+1}sum_{i=1}^{p+1}{(-1)^{delta_ip}{p+1choose i}}B_{p+1-i}n^{i}$ where $delta_ip$ is the Kronecker delta function and $B_{p+1-i}$ is the $p+1-i$th Bernoulli number.
My question is, what do I use for $i$ in the Kronecker delta function when using this formula?
For example, I am trying to derive the partial sum of the power series $sum_{k=1}^{n}{k^2}$ but I don't see what I would use for $i$.
calculus sequences-and-series summation binomial-coefficients
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
$endgroup$
$begingroup$
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
$endgroup$
– callculus
Jan 5 at 18:20
$begingroup$
Thank you @callculus. Yes, I will make that edit.
$endgroup$
– Gnumbertester
Jan 5 at 18:21
$begingroup$
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
$endgroup$
– Andy Walls
Jan 5 at 18:23
$begingroup$
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
$endgroup$
– Gnumbertester
Jan 5 at 18:25
1
$begingroup$
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
$endgroup$
– Andy Walls
Jan 5 at 18:40
add a comment |
$begingroup$
I am trying to use Faulhaber's formula to determine partial sums of a power series.
Faulhaber's formula is given by
$sum_{k=1}^{n}{k^{p}} = frac{1}{p+1}sum_{i=1}^{p+1}{(-1)^{delta_ip}{p+1choose i}}B_{p+1-i}n^{i}$ where $delta_ip$ is the Kronecker delta function and $B_{p+1-i}$ is the $p+1-i$th Bernoulli number.
My question is, what do I use for $i$ in the Kronecker delta function when using this formula?
For example, I am trying to derive the partial sum of the power series $sum_{k=1}^{n}{k^2}$ but I don't see what I would use for $i$.
calculus sequences-and-series summation binomial-coefficients
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
$endgroup$
I am trying to use Faulhaber's formula to determine partial sums of a power series.
Faulhaber's formula is given by
$sum_{k=1}^{n}{k^{p}} = frac{1}{p+1}sum_{i=1}^{p+1}{(-1)^{delta_ip}{p+1choose i}}B_{p+1-i}n^{i}$ where $delta_ip$ is the Kronecker delta function and $B_{p+1-i}$ is the $p+1-i$th Bernoulli number.
My question is, what do I use for $i$ in the Kronecker delta function when using this formula?
For example, I am trying to derive the partial sum of the power series $sum_{k=1}^{n}{k^2}$ but I don't see what I would use for $i$.
calculus sequences-and-series summation binomial-coefficients
calculus sequences-and-series summation binomial-coefficients
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
edited Jan 5 at 18:22
Gnumbertester
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
asked Jan 5 at 18:04
GnumbertesterGnumbertester
670114
670114
$begingroup$
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
$endgroup$
– callculus
Jan 5 at 18:20
$begingroup$
Thank you @callculus. Yes, I will make that edit.
$endgroup$
– Gnumbertester
Jan 5 at 18:21
$begingroup$
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
$endgroup$
– Andy Walls
Jan 5 at 18:23
$begingroup$
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
$endgroup$
– Gnumbertester
Jan 5 at 18:25
1
$begingroup$
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
$endgroup$
– Andy Walls
Jan 5 at 18:40
add a comment |
$begingroup$
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
$endgroup$
– callculus
Jan 5 at 18:20
$begingroup$
Thank you @callculus. Yes, I will make that edit.
$endgroup$
– Gnumbertester
Jan 5 at 18:21
$begingroup$
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
$endgroup$
– Andy Walls
Jan 5 at 18:23
$begingroup$
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
$endgroup$
– Gnumbertester
Jan 5 at 18:25
1
$begingroup$
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
$endgroup$
– Andy Walls
Jan 5 at 18:40
$begingroup$
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
$endgroup$
– callculus
Jan 5 at 18:20
$begingroup$
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
$endgroup$
– callculus
Jan 5 at 18:20
$begingroup$
Thank you @callculus. Yes, I will make that edit.
$endgroup$
– Gnumbertester
Jan 5 at 18:21
$begingroup$
Thank you @callculus. Yes, I will make that edit.
$endgroup$
– Gnumbertester
Jan 5 at 18:21
$begingroup$
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
$endgroup$
– Andy Walls
Jan 5 at 18:23
$begingroup$
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
$endgroup$
– Andy Walls
Jan 5 at 18:23
$begingroup$
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
$endgroup$
– Gnumbertester
Jan 5 at 18:25
$begingroup$
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
$endgroup$
– Gnumbertester
Jan 5 at 18:25
1
1
$begingroup$
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
$endgroup$
– Andy Walls
Jan 5 at 18:40
$begingroup$
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
$endgroup$
– Andy Walls
Jan 5 at 18:40
add a comment |
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$begingroup$
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
$endgroup$
– callculus
Jan 5 at 18:20
$begingroup$
Thank you @callculus. Yes, I will make that edit.
$endgroup$
– Gnumbertester
Jan 5 at 18:21
$begingroup$
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
$endgroup$
– Andy Walls
Jan 5 at 18:23
$begingroup$
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
$endgroup$
– Gnumbertester
Jan 5 at 18:25
1
$begingroup$
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
$endgroup$
– Andy Walls
Jan 5 at 18:40