# Changing variable in summation

• Jhenrique
In summary, there is a formula for computing summation by parts in integration, but it is also possible to make a change of variable in summation. However, the formula provided for the change of variable may not be correct. The correct formula is $$\sum_{i=1}^N f(u(x_i))\Delta x = \sum_{i=1}^N g(u_i)\Delta u : u_i=u(x_i)$$ and $$g(u_i) = f(x_i)\frac{\Delta x}{\Delta u} : x_i = h(u_i)=u^{-1}(u_i)$$ where ##f(u(x))dx \rightarrow f(u)(dx/du)du##
Jhenrique
Like in the integration, exist a formula to compute the summation by parts, that is: $$\frac{\Delta }{\Delta x}(f(x)g(x))=\frac{\Delta f}{\Delta x}g+f\frac{\Delta g}{\Delta x}+\frac{\Delta f}{\Delta x}\frac{\Delta g}{\Delta x}$$$$\sum \frac{\Delta }{\Delta x}(f(x)g(x))\Delta x = \sum \frac{\Delta f}{\Delta x}g\Delta x + \sum f\frac{\Delta g}{\Delta x}\Delta x + \sum \frac{\Delta f}{\Delta x}\frac{\Delta g}{\Delta x}\Delta x$$$$\sum f(x) \frac{\Delta g}{\Delta x}(x)\Delta x = f g - \sum \frac{\Delta f}{\Delta x} g \Delta x - \sum \frac{\Delta f}{\Delta x} \frac{\Delta g}{\Delta x}\Delta x$$http://en.wikipedia.org/wiki/Indefinite_sum#Summation_by_parts

But in the integration is possbile make a change the variable (integration by substitution). So, analogously, is possible to make a change the variable in the summation too?

I tried some like:$$\sum f(x) \Delta x =\sum f(x(u)) \frac{\Delta x}{\Delta u} \Delta u$$But, I think that this identity is wrong, because my calculus are wrong using this formula.

I don't think you've done the substitution correctly.

If you want:
$$\sum_{i=1}^N f(u(x_i))\Delta x = \sum_{i=1}^N g(u_i)\Delta u : u_i=u(x_i)$$

Then you want $$g(u_i) = f(x_i)\frac{\Delta x}{\Delta u} : x_i = h(u_i)=u^{-1}(u_i)$$

i.e. ##f(u(x))dx \rightarrow f(u)(dx/du)du##

## What is a changing variable in summation?

A changing variable in summation is a variable that is being iteratively added to a summation. It is usually represented by the letter "n" and is used to denote the number of terms being added in a sum.

## Why is it important to understand changing variables in summation?

Understanding changing variables in summation is important because it allows us to generalize mathematical expressions and find patterns in sequences and series. It also plays a crucial role in calculus and other branches of mathematics.

## How do you change the variable in a summation?

To change the variable in a summation, you can simply replace the original variable with the new variable. For example, if the original summation is written as ∑i=1n xi, and you want to change the variable from "i" to "j", the new summation would be ∑j=1n xj.

## What is the difference between a changing variable and a constant variable in summation?

A changing variable in summation varies with each term being added, while a constant variable remains the same throughout the summation. For example, in the summation ∑i=1n xi, "i" is the changing variable and "n" is the constant variable.

## How can changing variables in summation be used in real-life applications?

Changing variables in summation can be used in various real-life applications, such as calculating compound interest, estimating population growth, and finding the total cost of items with changing prices. It can also be used in computer programming and data analysis to process large amounts of data efficiently.

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