Linked Questions

27
votes
2answers
132k views

How to get to the formula for the sum of squares of first n numbers? [duplicate]

Possible Duplicate: How do I come up with a function to count a pyramid of apples? Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$? Finite Sum of Power? I know that the sum of ...
6
votes
4answers
4k views

Formula for partial sum of Riemann zeta function [duplicate]

Possible Duplicate: Finite Sum of Power? Suppose $f(s,k) = \sum_{n=1}^k n^{-s}$ is the Riemann zeta function truncated at the k-th term. I read on mathoverflow that there is a formula for $f(s,k)$...
9
votes
4answers
276 views

Is there a pattern to expression for the nested sums of the first $n$ terms of an expression? [duplicate]

Apologies for the confusing title but I couldn't think of a better way to phrase it. What I'm talking about is this: $$ \sum_{i=1}^n \;i = \frac{1}{2}n \left(n+1\right)$$ $$ \sum_{i=1}^n \; \frac{1}{...
2
votes
1answer
6k views

How to calculate $ 1^k+2^k+3^k+\cdots+N^k $ with given values of $N$ and $k$? [duplicate]

Here $ 1<N<10^9$ and $0<k<50$ So we have to calculate it in order of $O(\log N)$.
1
vote
4answers
441 views

Summation of n-squared, cubed, etc. [duplicate]

How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
2
votes
2answers
195 views

Evaluating $\sum_{k=1}^n k^x$ [duplicate]

Possible Duplicate: Finite Sum of Power? Is there a general expression for $\sum_{k=1}^n k^x$ for any integer value of $x$? The table for $x=1,2,\dots 10$ is given here. Is there formula for any ...
0
votes
2answers
112 views

How can i find summation of the series $i^k$ [duplicate]

Series : $$\sum_{i =1}^{n} i^k= 1^k+ 2^k + 3^k + 4^k +\ldots+n^k$$ where $k$ is a constant. This does not seem to be Geometric progression , how can I evaluate the sum? If possible if also want to ...
28
votes
13answers
4k views

Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula

As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer. In an attempt to make more interesting ...
0
votes
2answers
5k views

Sum of perfect squares to infinity

In this article, the method of computing $1+2+\dots$ is outlined. Is there a similar method for computing $1^2+2^2+\dots$? What about for the general power $n$? (That is, $1^n+2^n+\dots$)
0
votes
2answers
2k views

How to calculate the sum of $(n-1)^2+(n-2)^2+…+1$? [closed]

How to calculate the sum of the following series? $$(n-1)^2+(n-2)^2+...+1$$Thank you in advance
3
votes
4answers
1k views

how can one find the value of the expression, $(1^2+2^2+3^2+\cdots+n^2)$ [duplicate]

Possible Duplicate: Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$? Summation of natural number set with power of $m$ How to get to the formula for the sum of squares of first n ...
11
votes
2answers
388 views

Has anyone noticed this pattern?

I've been messing around a bit and I noticed a curious pattern when it comes to progressions of powers. Let's take the progression of consecutive integers: $1,2,3,4,5,6,7,...$ Obviously it's an ...
1
vote
3answers
90 views

Basic Mathematical Induction [duplicate]

I'm not quite sure how to approach this question. I need to prove that for $$n\ge1$$ $$1^2+2^2+3^3+\dots+n^2=\frac16n(n+1)(2n+1)$$ Do I just plug $1$ and see if $$\frac16(1)((1)+1)(2(1)+1) = 1^2\...
1
vote
1answer
116 views

Closed forms of sums of rational or real powers

I am curious about whether a closed form expression of $$\sum_{k=1}^{n}k^\alpha $$ for $\alpha \in \mathbb{R}$ exists in terms of special functions. Clearly for the natural number case we have the ...
1
vote
0answers
82 views

Sum of powers Anomaly!

Among other similarities, why is that in every $S_{ a }=\sum _{ j=1 }^{ n }{ { j }^{ a } }$ the denominator is the product of the sum of the coefficients of powers of n of individual factors? $ S_{ 1 ...