Problem 66 - Precalculus: Mathematical for Calculus

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 Pattern

Observe the sequence: \( 1, -2x, 3x^2, -4x^3, 5x^4, \ldots, -100x^{99} \). We see that the coefficients are alternating signs (+, -, +, -, ...) and are consecutive integers starting from 1 up to 100.

Step 2: Determine the General Term

Each term can be expressed as \((-1)^{n+1}nx^{n-1}\), where \(n\) is the coefficient of \(x^{n-1}\). This results in the signs alternating because \((-1)^{n+1}\) alternates between positive and negative as \(n\) increases.

Step 3: Set Up the Sigma Notation

The sequence starts from \(n=1\) to \(n=100\). Hence, the series can be expressed in sigma notation as: \[\sum_{n=1}^{100} (-1)^{n+1} n x^{n-1}\] This notation captures the entire sequence neatly.

Key Concepts

Alternating Series

An alternating series is a type of series where the signs of the terms alternate between positive and negative. In the given sequence, we see this pattern in action with the signs "+" and "-" switching back and forth every term. This is achieved by including a factor of \((-1)^{n+1}\) in the general term. \((-1)\) raised to an odd power remains negative, while raised to an even power, it becomes positive. This mathematical trick is what ensures that the series alternates its signs consistently. Alternating series are special because even if their terms don't decrease steadily to zero, they can still converge given certain conditions. An important feature is that alternating series can reflect complex functions and shapes simply by the nature of their pattern of opposites.

General Term

The general term of a sequence is a formula that lets you find any term in the sequence, without having to manually list all preceding terms. For our sequence, the general term is \((-1)^{n+1}nx^{n-1}\). Here, \(n\) represents our position in the sequence, starting from 1, and \(x^{n-1}\) indicates the variable part of the term, where the power of \(x\) increases with each step. This formula captures two important details:

• The alternating sign through \((-1)^{n+1}\), guaranteeing every other term switches signs.
• The integer sequence found in \(n\), which grows by 1 each time, starting from 1.

Together, these components precisely describe how to construct the sequence from individual pieces.

Sequence Pattern

Recognizing a sequence pattern is vital in identifying how a sequence progresses. The sequence we are looking at forms a pattern by increasing its power on \(x\) while alternating the signs. Recognizing the pattern helps us determine the sequence's behavior over time. For instance, the first few terms \(1, -2x, 3x^2, \ldots, -100x^{99}\) suggest several features:

• The terms alternate in sign due to the \((-1)^{n+1}\) term.
• The numeric sequence starts at 1 and increases linearly by 1.
• The exponent of \(x\) follows the numeric pattern in incrementing degrees.

A good grasp of pattern identification assists in predicting future terms or solving for missing terms in longer sequences.

Polynomial Series

A polynomial series is a sequence where each term is a polynomial. In our example, the terms like \(1, -2x, 3x^2,...\) fit the polynomial form \(ax^n\), where \(a\) is the coefficient and \(n\) determines the power of \(x\). Each additional term increases the exponent of \(x\) by 1, following a structured format quite typical for polynomials.This series is an example of a polynomial series that also features alternating signs, which makes it particularly interesting. Such series are often seen in various calculus contexts where they may approximate functions or describe certain behaviors like the convergence or divergence of the series.