Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

17770 questions
0
votes
1 answer

Can a fraction in a sum equation be flipped?

Given the equation: $$\sum_i^n \frac{1}{f(i)} \geq \sum_i^n \frac{1}{g(i)} $$ It looks intuitive to be able to flip the fraction and the sign, such that: $$\sum_i^n f(i) \leq \sum_i^n g(i)$$ Is this true? If so, how to prove it?
user15116257
  • 19
0
votes
3 answers

Can summation of product be evaluated by using the (a+b)^2 identity

Summation of $ij,$ can be evaluated by $\frac{(i+j)^2-i^2-j^2}{2}$ right?
Nikhil R
  • 17
0
votes
1 answer

Summation for powers of 2

I have a summation series of the form: n + (n-2^1) + (n - 2^2) + (n - 2^3) .... + (n - 2^l), where n is the total number of nodes in a tree, and l is the height of the tree. Is there a formula for this series?
John Doe
  • 3
0
votes
2 answers

Simplifying the summation $\sum_{j=0}^{n-2} (n - j)$

I was studying the iteration method and one sample had this summation: $\sum_{j=0}^{n-2} (n - j)$ Where this eventually gets simplified as: $n(n-1) - {(n-2)(n-1)\over 2} $ I did not quite understand how this is achieved. The first step is obviously…
Sarp Kaya
  • 183
0
votes
1 answer

Sum of the series $\sum_i |i|^\alpha$

I am looking for the sum to the following finite series: $ \sum_{i=0}^N |i|^\alpha $ where $0<\alpha<1$. Thanks.
yoki
  • 1,431
0
votes
0 answers

What is the general identity of this summation:

$$\sum_{i=1}^n n$$ I am confused by the solution to this question: $$\sum_{x=0}^{n-2} 4n=4n(n-1)$$ I know by the distributive property, the 4 can be moved in front of the summation. However, plugging $k=(n-2)$ in $k(k+1)/2$ doesn't return the…
Boy
  • 25
0
votes
1 answer

How to estimate the correction to the integral approximation of a discrete sum?

In the following approximation, $~\epsilon\sum_{n\in Z} F(\epsilon n)=\int_{-\infty}^{+\infty} dx~ F(x) + $ correction, how can one estimate the `leading' order correction for small $\epsilon$? Here the function, $F(x)$, and its all derivatives, …
Jeong-Hyuck Park
  • 91
0
votes
0 answers

Sum with $2$ variable $i$ and $j$

Calculation of $\displaystyle\mathop{\sum\sum}_{1\leq i< j\leq n}ij$ When I expand that sum, I get $\displaystyle[1(2+3+4+\cdots+n)+2(3+4+\cdots+n) $ $+3(4+5+6+\cdots+n)+\cdots (n-1)n]$ In each bracket sum in arthematic progression and we have to…
Priti Bisht
  • 421
0
votes
0 answers

What are the reasons that Abel summable does not imply Borel Summable?

I've been looking into summation regularizations, and I'm genuinely confused on why something that is Abel summable is not Borel summable? They both seem to be the same formula, then what is the difference?
Tsar Asterov XVII
  • 229
0
votes
0 answers

Define a summation of a set that consists of only non-numeric elements

Say I want to produce a sum of variable values. But I want the running index, say i, to refer to elements of a set of string, like $$M=\{{Banana, Bread, Computer}\}$$ such that $$i\in M$$ and that the result somehow resembles $$\sum_{i\in M}{}x_i$$…
RibbitRibbit
  • 1
0
votes
1 answer

Pattern in Dirichlet regularizations of divergent series.

So, I was trying to find a general pattern/formula for the Dirichlet Regularization of a Divergent series. I hadn't faced any problems until I reached: $$\sum_{n=0}^\infty n^x$$ So, let's make a function $\operatorname{Di}$, which basically gives…
Tsar Asterov XVII
  • 229
0
votes
0 answers

How to prove trig partial sum to -1

I was struggling with finding a proof to prove: $$ \sum_{0\leq n \leq \alpha -1} \cos{\frac{2n\pi}{\alpha}} = -1 $$ I know it's true but I cannot figure out how to prove it, Sorry to ask such a simple question, And many thanks to whoever can help.
Adrian Salt
  • 51
0
votes
0 answers

How to derive the formula for the sum of the first $n$ natural numbers?

I am doing some questions on: Obtaining the formula for the sum of the first $n$ odd natural numbers which I have got as $n^2$ Obtaining the formula for the sum of the first $n$ even natural numbers which I have got as $n(n+1)$ I am now supposed…
aelcro
  • 9
0
votes
2 answers

How to convert from a sum written out long-hand to sigma notation

We have $\color{green}{a.}$ $$\left(\frac{b}{n}\right)\left(\frac{b}{n}\right)^2+\left(\frac{b}{n}\right)\left(\frac{2b}{n}\right)^2+\left(\frac{b}{n}\right)\left(\frac{3b}{n}\right)^2+\cdots+\left(\frac{b}{n}\right)\left(\frac{nb}{n}\right)^2…
Haizum Skallah
  • 653
0
votes
0 answers

For a sequence of values, how can you calculate the index of X?

I first thought this would be easy but I can't think of a solution. Say you have a list of numbers with values between 0 and 1. I need a way to calculate what is the index of the interval in which X is. Here is a demonstration of what i'm trying to…
Samuel Fyckes
  • 119
1 2 3
…
99 100