Problem 66 - Algebra and Trigonometry

Question

Write the sum using sigma notation. \(1-2 x+3 x^{2}-4 x^{3}+5 x^{4}+\cdots-100 x^{99}\)

Step-by-Step Solution

Answer

\( \sum_{n=1}^{100} (-1)^{n+1} n x^{n-1} \)

Step 1: Identify the Sequence

Observe that the given series starts at 1 and proceeds with alternating positive and negative terms. The series is: \[ 1 - 2x + 3x^2 - 4x^3 + 5x^4 + ext{...} - 100x^{99} \] This indicates an alternating sign structure based on the power of \(x\).

Step 2: Determine the General Term

The general term follows the pattern of alternating terms with increasing powers of \(x\) and coefficients increasing linearly. The positive integers \(n\) are increasing as coefficients and the power of \(x\) is increasing by 1 each time. Considering these, the general term formula is:\[ (-1)^{n+1} n x^{n-1} \] where \(n\) starts from 1.

Step 3: Express the Sum in Sigma Notation

The series can now be rewritten in sigma notation using the general term we derived. Since the series starts from the first term, \(n = 1\), and ends at the 100th term, \(n = 100\), the sum is:\[ \sum_{n=1}^{100} (-1)^{n+1} n x^{n-1} \]

Key Concepts

Alternating Series

In mathematics, an alternating series is a sequence of numbers in which the signs alternate all the time. This means that if one term is positive, the next will be negative and vice versa. In our exercise, we see an alternating pattern in the series:

• It starts with a positive number, 1, and switches its sign each following term.
• The negative term appears at every odd power of x and the positive term appears at every even power.

So, the series takes the form of alternating between positive and negative terms based on its position. This can be useful in adding variability and complexity to the series structure. Alternating series are widely used for various mathematical and practical applications because they naturally introduce changes in direction at predictable intervals.

General Term

Identifying the general term of a series is crucial in working with complicated expressions. It allows us to describe any term in the series using a formula. In our series, the formula for the general term is:

• The sign alternates based on the term's position:\[ (-1)^{n+1} \]
• The coefficients are increasing by one at each step, represented by:\[ n \]
• The power of x also increases as the term progresses, denoted by:\[ x^{n-1} \]

This general term, \[ (-1)^{n+1} nx^{n-1} \], effectively encapsulates the behavior and structure of the sequence, making it easy to express any term without listing them all.

Series Expression

When expressing a series, capturing the overall pattern in a concise way is important. This is where sigma notation comes in. It allows you to express an entire series succinctly.Our series:\[ 1 - 2x + 3x^2 - 4x^3 + 5x^4 + ... - 100x^{99} \]can be written in sigma notation as:\[ \sum_{n=1}^{100} (-1)^{n+1} nx^{n-1} \]Here, the expression:

• Starts with \( n = 1 \) and ends at \( n = 100 \) because these are the limits of the terms we have.
• Uses the general term we identified to calculate each element in the series.

Sigma notation offers a powerful way to represent series, especially those with many terms, without needing to write out each term individually.

Algebraic Series

Algebraic series involve summing terms that are defined by algebraic expressions. In our context, the series involves a polynomial pattern where the expression is related to a power of a variable, like \( x \). Some characteristics include:

• It has coefficients that are changing predictably (increasing by one).
• It includes a variable \( x \) raised to the power that directly relates to the term's position.

The algebraic series can often be found in numerous mathematical problems and helps in modeling real-life scenarios where relationships are defined with polynomial functions. Understanding these makes the interpretation of the series straightforward, enabling better analysis and application.