Linked Questions

1
vote
2answers
1k views

The sum of the fourth powers of the first $n$ positive integers [duplicate]

I am studying mathematical induction and most of the times I have to prove something. Like, for example: $1 + 4 + 9 + ...+ n^2 = \frac{n(n+1)(2n+1)}{6}$ This time I found a question that ask me to ...
3
votes
2answers
104 views

How to prove that $1^r + 2^r +… + n^r = a_1 n^1 + a_2 n^2 +… + a_{r+1} n^{r+1}$? [duplicate]

I see this equation $$ 1^r + 2^r +... + n^r = a_1 n^1 + a_2 n^2 +... + a_{r+1} n^{r+1} $$ in Introduction to Linear Algebra, where $a_k$s are some constants. How can I prove it?
0
votes
0answers
41 views

What is the sum of the given special series?any specific formula? [duplicate]

$1^n$+$2^n$+$3^n$+$\dots$+$m^n$. $(m,n\in\mathcal N)$ $m,n\ge2$
120
votes
31answers
61k views

Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$

I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ...
37
votes
17answers
4k views

I'm looking for some mathematics that will challenge me as a year $12$ student. [closed]

I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions. I want a challenge, ...
11
votes
14answers
505 views

Different ways to come up with $1+2+3+\cdots +n=\frac{n(n+1)}{2}$

I am trying to compile a list of the different ways to come up with the closed form for the sum in the title. So far I have the famous Gauss story, the argument by counting doubletons, and using ...
7
votes
12answers
2k views

Compilation of proofs for the summation of natural squares and cubes

I want to know different proofs for the following formulas, $$ \sum_{i=1}^n{i^2} = \frac{(n)(n+1)(2n+1)}{6} $$ $$ \sum_{i=1}^n{i^3} = \frac{n^2(n+1)^2}{2^2} $$ Please do not mark this as duplicate, ...
23
votes
5answers
12k views

A formula for the power sums: $1^n+2^n+\dotsc +k^n=\,$?

Is there explicit formula for the expression $1^n + 2^n + \dotsc + k^n\,$? I know that for $n=1$ the explicit formula becomes $S=k(k+1)/2$ and for $n=3$ the formula becomes $S^2$. But what about ...
9
votes
5answers
1k views

How Are the Solutions for Finite Sums of Natural Numbers Derived?

So, I've been learning set theory on my own (Lin, Shwu-Yeng T., and You-Feng Lin. Set Theory: An Intuitive Approach. Houghton Mifflin Co., 1974.) and have come across infinite sums of natural numbers. ...
14
votes
2answers
4k views

Finite Sum of Power?

Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^p$$ For $p = 1$, it's just the classic $\frac{n(n+1)}2$. What is it for $p > 1$?
8
votes
3answers
996 views

Closed form for $1^k + … + n^k$ (generalized Harmonic number)

This question must have been asked, it's just very hard to search for such questions. I'm looking for the cleanest method I can find for getting a closed form formula for $\sum_{i=1}^n i^k$ ...
8
votes
3answers
1k views

Formula for $1^k+2^k+3^k…n^k$ for $n,k \in \mathbb{N}$

So I've been looking for a formula where I can input the parameter $k$ and it will give me a formula for $1^k+2^k+3^k...+ n^k$ with $n,k \in \mathbb{N}$. The result is always a polynomial with $k+1$ ...
7
votes
2answers
744 views

The sum of fractional powers $\sum\limits_{k=1}^x k^t$.

This post is a continuation of Generalization of the Bernoulli polynomials ( in relation to the Index ) , the definition of the Bernoulli polynomial $B_t(x)$ with $|x|<1$ has an extension through $...
1
vote
3answers
112 views

A formula for $1^4+2^4+…+n^4$

I know that $$\sum^n_{i=1}i^2=\frac{1}{6}n(n+1)(2n+1)$$ and $$\sum^n_{i=1}i^3=\left(\sum^n_{i=1}i\right)^2.$$ Here is the question: is there a formula for $$\sum^n_{i=1}i^4.$$
2
votes
2answers
311 views

How to obtain a closed form for summation over polynomial ($\sum_{x=1}^n x^m$)? [duplicate]

What is the method for obtaining the polynomial equal to \begin{equation*} \sum^{n}_{x=1}x^m \end{equation*} for unknown $n$, and systematically for various values of $m$? I know it should be a ...

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