Understanding Kronecker Delta Function for Faulhaber's Formula
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 2 days ago
GnumbertesterGnumbertester
1285
add a comment |
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 2 days ago
GnumbertesterGnumbertester
1285
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
– callculus
2 days ago
Thank you @callculus. Yes, I will make that edit.
– Gnumbertester
2 days ago
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
– Andy Walls
2 days ago
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
– Gnumbertester
2 days ago
1
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
– Andy Walls
2 days ago
add a comment |
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 2 days ago
GnumbertesterGnumbertester
1285
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 2 days ago
GnumbertesterGnumbertester
1285
asked 2 days ago
GnumbertesterGnumbertester
1285
edited 2 days ago
Gnumbertester
asked 2 days ago
GnumbertesterGnumbertester
1285
asked 2 days ago
GnumbertesterGnumbertester
1285
asked 2 days ago
GnumbertesterGnumbertester
1285
1285
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
– callculus
2 days ago
Thank you @callculus. Yes, I will make that edit.
– Gnumbertester
2 days ago
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
– Andy Walls
2 days ago
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
– Gnumbertester
2 days ago
1
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
– Andy Walls
2 days ago
add a comment |
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
– callculus
2 days ago
Thank you @callculus. Yes, I will make that edit.
– Gnumbertester
2 days ago
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
– Andy Walls
2 days ago
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
– Gnumbertester
2 days ago
1
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
– Andy Walls
2 days ago
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
– callculus
2 days ago
You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
– callculus
2 days ago
Thank you @callculus. Yes, I will make that edit.
– Gnumbertester
2 days ago
Thank you @callculus. Yes, I will make that edit.
– Gnumbertester
2 days ago
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
– Andy Walls
2 days ago
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
– Andy Walls
2 days ago
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
– Gnumbertester
2 days ago
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
– Gnumbertester
2 days ago
1
1
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
– Andy Walls
2 days ago
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
– Andy Walls
2 days ago
add a comment |
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You mean $sumlimits_{k=1}^n color{red}k^2=frac16cdot ncdot (n+1)cdot (2n+1)$?
– callculus
2 days ago
Thank you @callculus. Yes, I will make that edit.
– Gnumbertester
2 days ago
$delta_{ip} = 1 space mathrm{for} space i=p$ and 0 otherwise. So one term gets a multiplier of $-1$ while all the others just get a multiplier of $1$.
– Andy Walls
2 days ago
@AndyWalls , yes, I understand how to evaluate $delta_{ip}$. I just don't understand what $i$ is in the context of Faulhaber's formula.
– Gnumbertester
2 days ago
1
It is the index of the summation on the RHS, i.e. the index of each term in the summation.
– Andy Walls
2 days ago