Finding the sum of the first n fifth powers

[burt]

I am trying to write out the sum of the first n fifth powers, using Bernoulli's formula. I know that the correct answer is \(\frac{1}{6}n^6+\frac{1}{2}n^5+\frac{5}{12}n^4-\frac{1}{12}n^2\). However, I also got a term of \(\frac{1}{252}\), without any n. Is my problem that you are only supposed to take the formula as far as you will have n to the positive power - without including n^0?

 

[Dr.Peterson]

I am trying to write out the sum of the first n fifth powers, using Bernoulli's formula. I know that the correct answer is \(\frac{1}{6}n^6+\frac{1}{2}n^5+\frac{5}{12}n^4-\frac{1}{12}n^2\). However, I also got a term of \(\frac{1}{252}\), without any n. Is my problem that you are only supposed to take the formula as far as you will have n to the positive power - without including n^0?

Please show us Bernoulli's formula in the form you were taught. This may just be a matter of reading the formula carefully.

 

[burt]

Please show us Bernoulli's formula in the form you were taught. This may just be a matter of reading the formula carefully.

\(\frac{1}{c+1}n^{c+1}+\frac12n^c+\frac{c}{2}B_2n^{c-1}+\frac{c(c-1)(c-2)}{2\bullet3\bullet4}B_4n^{c-3}+\frac{c(c-1)(c-2)(c-3)(c-4)}{2\bullet3\bullet4\bullet5\bullet6}B_6n^{c-5}+...\) where \(B_2, B_4...\) are the Bernoulli numbers.

 

[Dr.Peterson]

\(\frac{1}{c+1}n^{c+1}+\frac12n^c+\frac{c}{2}B_2n^{c-1}+\frac{c(c-1)(c-2)}{2\cdot3\cdot4}B_4n^{c-3}+\frac{c(c-1)(c-2)(c-3)(c-4)}{2\cdot3\cdot4\cdot5\cdot6}B_6n^{c-5}+...\) where \(B_2, B_4...\) are the Bernoulli numbers.

I am not familiar with the formula, so I can't fully explain this, but it does appear that your version of the formula doesn't explicitly tell you where to stop. The formula I had found, here and here, is



This explicitly tells you to stop when k=p, so the final exponent is p-p+1=1, not 0.

I found essentially your version here, but nothing I've seen explains how you are supposed to know this.

 

[burt]

I am not familiar with the formula, so I can't fully explain this, but it does appear that your version of the formula doesn't explicitly tell you where to stop. The formula I had found, here and here, is

View attachment 24957

This explicitly tells you to stop when k=p, so the final exponent is p-p+1=1, not 0.

I found essentially your version here, but nothing I've seen explains how you are supposed to know this.

So it seems that I should stop when the final exponent is 1. That would make sense, as I know the correct answer in this case.