Problem 39 - Algebra for College Students

Question

Write out the terms of each series. $$\sum_{i=1}^{3} i x^{i}$$

Step-by-Step Solution

Answer

The terms are \(x\), \2x^2\), and \3x^3\).

Step 1 - Understand the Summation Notation

The notation \(\sum_{i=1}^{3} i x^{i}\) indicates that we need to sum the expression \(i x^{i}\) for values of \(i\) going from 1 to 3.

Step 2 - Substitute the Values of i

Start by substituting \(i = 1\): \(1 x^{1} = x\). \Next, substitute \(i = 2\): \2 x^{2}\. Finally, substitute \(i = 3\): \3 x^{3}\.

Step 3 - Write Out the Terms

Combine all the terms together resulting from the substitution: \(x + 2x^2 + 3x^3\).

Key Concepts

Series

A series is the sum of the terms of a sequence. In this exercise, we are dealing with the series \(\sum_{i=1}^{3} i x^{i}\). This notation tells us to sum up the terms generated by substituting the values of \(\i\) from 1 to 3 into the expression \(i x^{i}\). Series can be finite or infinite. In our case, it's finite because it runs from 1 to 3. Each value of \(\i\) gives us a term, resulting in a sum of all these terms.

Substitution

Substitution is the process of replacing a variable with its corresponding value. In this problem, we substitute values of \(\i\) into the expression \(i x^{i}\). Here's how we do it:
- For \(\i=1\): \(1 x^{1} = x\)
- For \(\i=2\): \(2 x^{2}\)
- For \(\i=3\): \(3 x^{3}\)

This process helps us determine each individual term of the series before summing them up.

Algebraic Expressions

An algebraic expression consists of variables, constants, and operations like addition and multiplication. In this exercise, the expression \(i x^{i}\) consists of the variable \(x\) and the coefficient \(i\) which changes with each step in the summation notation. So, every term has the form \(i x^{i}\), where \(i\) varies from 1 to 3. This helps in creating more complex expressions which can be simplified or expanded.

Summation

Summation notation, also known as sigma notation, is a concise way of writing the sum of a sequence. The \(\sum\) symbol (sigma) tells us to add up all terms specified by the expression following it. In this exercise:\