# How Can Nested Summation Functions Be Simplified or Reversed?

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• ReubenDS
The topic of finding the roots of polynomial equations can be found in the subject of "numerical analysis".
ReubenDS
TL;DR Summary
Summation of Summation method
Extracting value for n of a Summation
1. I have come across a few times I would like a more straightforward way to run a summation function on a summation function. I don't have the educational groundwork to know if there is another way to do this or a good technique to simplify these problems.
example:
2*Sum[2x-1*-Sum[Divide[2x-1,2],{x,1,n}],{x,1,n}]
Where 2* summation of all whole integers of the function (2x-1)*-summation of all whole integers of the function (2x-1)/2 where x=1 to n for both summations.

2. It has struck me from time to time that I would like to reverse a summation function to derive an unknown limit for a known resultant of the summation.
example:
(Sum[2x-1,{x,1,n}])=2704
Where the summation of all whole integers of the function 2x-1 for x=1 to n, equals 2704 what is the value of n?

Example 1
$2 \sum_{x=1}^n 2x-1- \sum{x=1}^n \frac {2x-1} 2$

Example 2
$\sum_{x=1}^n 2x-1 = 2704$

**I do not care for the answers to these example problems

Last edited by a moderator:
ReubenDS said:
**I do not care for the answers to these example problems
What do you care for? I have difficulties understanding your question. Do you want to know how to read ##\sum_{k=1}^n a_k## or how to program a sum?

I've formatted it in Latex, and I can't explain why it doesn't display correctly.

The question is about proper notations I should be using or techniques to simplify down to my current notation.

If we want to sum up an expression, say ##f(k)## that depends on a natural number ##k## from some lower value ##L## of ##k## to some upper value ##U## of ##k##. That is
$$f(L)+f(L+1)+f(L+2)+\ldots+f(U-2)+f(U-1)+f(U)=\sum_{k=L}^U f(k)$$
For example ##1+2+3+\ldots+n=\displaystyle{\sum_{k=1}^n k} =\dfrac{n(n+1)}{2}## or
$$9+16+25+\ldots+n^2=\sum_{k=3}^n k^2=\left(\sum_{k=1}^n k^2\right)-4-1=\dfrac{n(n+1)(2n+1)}{6}-5$$

ReubenDS said:
Example 1
$2 \sum_{x=1}^n 2x-1- \sum{x=1}^n \frac {2x-1} 2$
What does the notation ##\sum x ## mean in this context? What are the limits on that summation?

ReubenDS said:
Example 2
$\sum_{x=1}^n 2x-1 = 2704$
I'll assume you are asking about the result of ##\sum_{x=1}^n P(x) ## where ##P(x)## is a polynomial in ##x## (such as P(x) = 2x -1 ). Such a sum can be expressed as ##Q(n)## where ##Q(n)## is a polynomial in ##n##. Apparently you are asking about solving ##Q(n) = c## where ##c## is a specified value. That's the problem of solving for the root(s) of polynomial ##Q(n) - c##.

Finding the function ##Q(n)## is covered in the mathematical topic of "The Calculus of Finite Differences".

The topic of finding the roots of polynomials is covered in elementary and advanced courses on algebra.

## 1. What is summation in mathematics?

Summation is a mathematical operation that involves adding together a sequence of numbers or terms. It is often denoted by the symbol Σ (sigma) and is used to find the total or sum of a set of values.

## 2. What is the purpose of summation?

The purpose of summation is to find the total or sum of a set of numbers or terms. It is commonly used in mathematics, statistics, and physics to calculate averages, probabilities, and other important quantities.

## 3. How do you perform summation?

To perform summation, you need to first identify the sequence of numbers or terms that you want to add together. Then, you can use the sigma notation (Σ) to represent the summation. Finally, you can use the appropriate formula or method to calculate the sum.

## 4. What is the difference between summation and integration?

Summation and integration are both mathematical operations that involve adding together a sequence of numbers or terms. However, summation is used for discrete values, while integration is used for continuous values. In other words, summation is used for adding together individual values, while integration is used for finding the area under a curve.

## 5. What are some common applications of summation?

Summation has many applications in various fields such as mathematics, statistics, physics, and computer science. Some common applications include calculating probabilities, finding the average or mean of a set of values, and determining the total distance or displacement traveled in physics problems.

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