Problem 65 - Precalculus: Mathematical for Calculus

Question

Write the sum using sigma notation. $$1+x+x^{2}+x^{3}+\cdots+x^{100}$$

Step-by-Step Solution

Answer

The sum is written as \( \sum_{n=0}^{100} x^n \).

Step 1: Identify the Pattern

Look at the sum: \(1, x, x^2, x^3, \ldots, x^{100}\). Notice that each term is a power of \(x\), starting from \(x^0\) and ending at \(x^{100}\). This sequence suggests a pattern of powers of \(x\).

Step 2: Define the General Term

From the pattern, we can see that the general term of the sequence is \(x^n\), where \(n\) starts from \(0\) and goes up to \(100\). Thus, the general term that we want to sum is \(x^n\).

Step 3: Write the Sum using Sigma Notation

Sigma notation is used to write sums compactly. The general form is \( \sum_{n=a}^{b} \, f(n) \), where \(a\) and \(b\) are the limits of the sum and \(f(n)\) is the general term. In this case, the sum becomes \( \sum_{n=0}^{100} x^n \).

Key Concepts

Power Series

Power series are expressions of the form \( a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots \). They allow us to represent functions as infinite sums through a sequence of coefficients \(a_n\). Power series are a powerful tool in mathematics because they can represent many important functions. For example, the exponential function \( e^x \) is represented by the power series \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \).

• Each term in the series is a multiple of a power of \(x\).
• The number of terms in a power series can be finite or infinite.
• The general form of a power series can be written as \( \sum_{n=0}^{\infty} a_n x^n \).

Understanding power series is essential for grasping many topics, such as convergence of series and functions' approximations. They are instrumental in calculus, helping to simplify complex functions into an easier series of calculations.

Geometric Series

A geometric series is a specific type of series where each term is a constant multiple of the previous one. If you look at the series \(1 + x + x^2 + x^3 + \ldots + x^{100}\), it is a geometric series because each term is obtained by multiplying the previous term by \(x\). The series ends with the term \(x^{100}\), making it a finite geometric series.

• In a geometric series, the ratio between successive terms is constant, often referred to as the common ratio.
• The formula for the sum of a finite geometric series is \( S_n = a \frac{1-r^n}{1-r} \), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
• For an infinite geometric series with \(|r| < 1\), the sum is \( \frac{a}{1-r} \).

Understanding geometric series is useful in various fields, like finance, physics, and computer science, where growth and decay processes can often be modeled using geometric relationships.

Summation Notation

Summation notation, often represented using the Greek letter sigma (\( \Sigma \)), is a method to denote the sum of a sequence of terms. Instead of writing out each term individually, summation notation provides a compact way to express series and sums using a clear mathematical format.

• The typical sigma notation form is \( \sum_{n=a}^{b} f(n) \), where \(n\) is the index of summation, \(a\) starts the sum, \(b\) ends it, and \(f(n)\) is the function defining the terms.
• It allows mathematicians to convey large sums succinctly, such as \( \sum_{n=0}^{100} x^n \) for the series \(1 + x + x^2 + \cdots + x^{100}\).
• This notation is versatile and can represent both finite and infinite sums depending on the context.

Exploring summation notation is crucial for mathematical proficiency, as it is widely used across disciplines in solving problems and expressing mathematical ideas efficiently.