Work Inspired by a wiki page:

This wiki page page inspired me to do discover i=1nia\sum^n_{i=1}i^a with aa being a natural number; I will just start using i=1ni0=10+20+30+...+n0=n\sum^n_{i=1}i^0=1^0+2^0+3^0+...+n^0=n so I can discover the next ones; Using the same method that the linked page had done with n2n^2 and n3n^3, I will discover the polynomial that is equal to i=1nia\sum^n_{i=1}i^a;

First, I will use binomial theorem:

(n+1)a+1=i=0a+1(a+1i)ni(n+1)^{a+1}=\sum^{a+1}_{i=0}\binom{a+1}{i}n^i The 1a+1i1^{a+1-i} will be always 1, so it will not change the result if not written

Therefore:

(n+1)a+1=na+1+i=0a(a+1i)ni(n+1)^{a+1}=n^{a+1}+\sum^a_{i=0}\binom{a+1}{i}n^i (I took the "na+1n^{a+1}" out of the sum notation)

(n+1)a+1na+1=i=0a(a+1i)ni(n+1)^{a+1}-n^{a+1}=\sum^a_{i=0}\binom{a+1}{i}n^i

As the page did, I will substitute n by 1,2,3....n

2a+11a+1=i=0a(a+1i)1i2^{a+1}-1^{a+1}=\sum^a_{i=0}\binom{a+1}{i}1^i

3a+12a+1=i=0a(a+1i)2i3^{a+1}-2^{a+1}=\sum^a_{i=0}\binom{a+1}{i}2^i

4a+13a+1=i=0a(a+1i)3i4^{a+1}-3^{a+1}=\sum^a_{i=0}\binom{a+1}{i}3^i

\vdots

na+1(n1)a+1=i=0a(a+1i)(n1)in^{a+1}-(n-1)^{a+1}=\sum^a_{i=0}\binom{a+1}{i}(n-1)^i

(n+1)a+1na+1=i=0a(a+1i)ni(n+1)^{a+1}-n^{a+1}=\sum^a_{i=0}\binom{a+1}{i}n^i

Then I will sum everything, notice that everything on the left side, except (n+1)a+1(n+1)^{a+1} and 1a+1-1^{a+1} cancels themselves, for the right side, every sum has now another sum inside it!

(n+1)a+11=i=0a((a+1i)j=1nji)(n+1)^{a+1}-1=\sum^a_{i=0}\left (\binom{a+1}{i}\sum^n_{j=1}j^i \right )

I will work the left side now, remember what (n+1)a+1(n+1)^{a+1} is, but now, we will have to take the (a+10)n0=1\binom{a+1}{0}n^0=1, because it will cancel with 1-1:

i=1a+1(a+1i)ni=i=0a((a+1i)j=1nji)\sum^{a+1}_{i=1}\binom{a+1}{i}n^i=\sum^a_{i=0}\left (\binom{a+1}{i}\sum^n_{j=1}j^i \right )

Now, I will take the term that appears on the sum of the right when i=a out of the notation:

i=1a+1(a+1i)ni=(a+1a)j=1nja+i=0a1((a+1i)j=1nji)\sum^{a+1}_{i=1}\binom{a+1}{i}n^i=\binom{a+1}{a}\sum^n_{j=1}j^a+\sum^{a-1}_{i=0}\left (\binom{a+1}{i}\sum^n_{j=1}j^i \right )

Notice that this term is exactally what we are searching, so I will isolate it:

(a+1a)j=1nja=i=1a+1(a+1i)nii=0a1((a+1i)j=1nji)\binom{a+1}{a}\sum^n_{j=1}j^a=\sum^{a+1}_{i=1}\binom{a+1}{i}n^i-\sum^{a-1}_{i=0}\left (\binom{a+1}{i}\sum^n_{j=1}j^i \right )

Remember that (a+1a)=(a+11)=a+1\binom{a+1}{a}=\binom{a+1}{1}=a+1:

j=1nja=i=1a+1(a+1i)nii=0a1((a+1i)j=1nji)a+1\sum^n_{j=1}j^a=\frac{\sum^{a+1}_{i=1}\binom{a+1}{i}n^i-\sum^{a-1}_{i=0}\left (\binom{a+1}{i}\sum^n_{j=1}j^i \right )}{a+1}

Now, Remember that, when a=0a=0, we have a polynomial expression nn, then if a=2a=2 also "is"(As in: there is a polynomial expression that has the same values) a polynomial expression, and, by a strong induction, if a is natural, then the expression i=1nia\sum^n_{i=1}i^a also "is".

I also made a program, in Python, that returned the expression of i=1nia\sum^n_{i=1}i^a, but to much more values other than 0, 1, 2, 3(Which the original page did), some of the results are:

i=1ni1=n2+n2\sum^n_{i=1}i^{1}=\frac{n^2 + n}{2}

i=1ni2=2n3+3n2+n6\sum^n_{i=1}i^{2}=\frac{2n^3 + 3n^2 + n}{6}

i=1ni3=n4+2n3+n24\sum^n_{i=1}i^{3}=\frac{n^4 + 2n^3 + n^{2}}{4}

i=1ni4=6n5+15n4+10n3n30\sum^n_{i=1}i^{4}=\frac{6n^5 + 15n^4 + 10n^3 - n}{30}

i=1ni5=2n6+6n5+5n41n212\sum^n_{i=1}i^{5}=\frac{2n^6 + 6n^5 + 5n^4 - 1n^2}{12}

i=1ni6=6n7+21n6+21n57n3+n42\sum^n_{i=1}i^{6}=\frac{6n^7 + 21n^6 + 21n^5 - 7n^3 + n}{42}

i=1ni7=3n8+12n7+14n67n4+2n224\sum^n_{i=1}i^{7}=\frac{3n^{8} + 12n^{7} + 14n^{6} - 7n^{4} + 2n^{2}}{24}

i=1ni8=10n9+45n8+60n742n5+20n33n90\sum^n_{i=1}i^{8}=\frac{10n^{9} + 45n^{8} + 60n^{7} - 42n^{5} + 20n^{3} - 3n}{90}

i=1ni9=2n10+10n9+15n814n6+10n43n220\sum^n_{i=1}i^{9}=\frac{2n^{10} + 10n^{9} + 15n^{8} - 14n^{6} + 10n^{4} - 3n^{2}}{20}

There are patterns there, but I will make another note to that, as this is getting huge

Note by Matheus Jahnke
2 years, 10 months ago

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This is amazing, i never knew we can generalize it !! Before reading this note , i used to approximate the answer.

Sk=1k+2k+3k++nk=r=1nrk S_{k} = 1^{k}+2^{k}+3^{k}+\cdots+n^{k} =\displaystyle\sum_{r=1}^{n} r^{k}

For a large value of n n (probably greater than 10001000 for negligible error) we can convert the sum in an integral

r=0nrk=nk+101xkdxSknk+1k+1 \displaystyle\sum_{r=0}^{n}r^{k} = n^{k+1} \int_{0}^{1} x^{k} dx \\ S_{k} \approx \dfrac{n^{k+1}}{k+1}

For example take k=1 k =1 and n=2000 n = 2000

S1200022=2000000exact sum = 2001000 S_{1} \approx \dfrac{2000^{2}}{2} = 2000000 \\ \text{exact sum = } 2001000

an error of only 1000 1000 . the bigger the value of n n , more accurate is the approximation

Sabhrant . - 2 years, 10 months ago

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That is one of the patterns:

The term with the highest degree of the equivalent of i=1nia\sum^n_{i=1}i^a is na+1a+1\frac{n^{a+1}}{a+1}

We can prove it by induction, as in: it works for a=1a = 1, because i=1ni1=n21+1+...\sum^n_{i=1} i^1=\frac{n^2}{1+1} + ...

If it works for a=k,k1,k2,...a = k, k-1,k-2, ..., then for a=k+1a = k + 1, we have:

i=1nik+1=i=1k+2(k+2i)ni(i=0a(k+2i)j=1nji)k+2\sum^n_{i=1}i^{k+1}=\frac{\sum^{k+2}_{i=1}\binom{k+2}{i}n^i-\left(\sum^{a}_{i=0}\binom{k+2}{i}\sum^n_{j=1}j^i\right )}{k+2}

The term with the highest degree is n(k+1)+1(k+1)+1\frac{n^{(k+1)+1}}{(k+1)+1} due the first sum highest degree, no term of the other sum has a equal or higher degree, because the highest one is i=1nik\sum^n_{i=1}i^k, which gives nk+1k+1\frac{n^{k+1}}{k+1}

And as the others sums have a lower exponent, then they will have a lower degree of term(this is a strong induction)

Matheus Jahnke - 2 years, 10 months ago

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There's a (somewhat) general result for this involving Bernoulli numbers. It's called Faulhaber's formula.

Prasun Biswas - 2 years, 10 months ago

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That is a very good comment, thanks for linking it, even though it make everything I did useless...

Matheus Jahnke - 2 years, 10 months ago

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