Question

What sort of big \(\Theta\) bound can you give on the value of a geometric series \(1+r+r^{2}+\cdots+r^{n}\), with common ratio \(r=1 ?\)

Short answer

The series has a big \( \Theta(n) \) bound.

Step-by-step solution

Identify the Pattern of the Sequence

The series in question is a geometric series: \(1 + r + r^2 + \cdots + r^n\). We are given that the common ratio \(r = 1\). When the common ratio is 1, every term in the series is the same, which alters the typical geometric series behavior.

Evaluate the Series with r = 1

Since \(r = 1\), each term of the series is simply 1. Therefore, the series becomes \(1 + 1 + 1 + \cdots + 1\) for a total of \(n+1\) terms, which sums up to \(n+1\).

Determine the Big Theta Notation

The sum of the series \(1 + 1 + \cdots + 1\) for \(n+1\) terms is a straightforward arithmetic sum. In Big Theta notation, the asymptotic behavior of this series is expressed as \( \Theta(n) \), because the series sum grows linearly with the number of terms \(n+1\). The constant +1 does not affect the asymptotic growth.

Key concepts

Big Theta Notation

Big Theta notation is used to describe the asymptotic behavior of functions. It gives a precise way of showing both the upper and lower bounds on the growth rate of a function. When we say a function is \(\Theta(g(n))\), it means that the function grows at the same rate as \(g(n)\), up to constant factors.
For example, if we have a function \(f(n) = 3n + 2\), we would say it's \(\Theta(n)\). Why? Because, as \(n\) becomes very large, the dominant term is \(3n\), which grows linearly with \(n\). The constant \(+2\) becomes insignificant in the long run, so it doesn't affect this notation.

• Purpose: To classify functions based on how they grow.
• Usage: Commonly used in computer science to analyze algorithms and determine their efficiency.
• Significance: Provides a balanced view with both lower and upper bounds, ensuring a tight description.

Asymptotic Analysis

Asymptotic analysis is a powerful technique that allows us to understand the behavior of algorithms as input size grows. It mainly focuses on the efficiency and performance of an algorithm when the input size is large.
In the context of a geometric series, asymptotic analysis helps us to grasp how the sum of the series behaves as the number of terms increases. For the series \(1+r+r^2+\dots+r^n\) with \(r=1\), the series simplifies to an arithmetic sum because each term is \(1\). Thus, the series becomes \(n+1\) terms of \(1\), and the analysis reveals that its growth is linear. This is shown by the \(\Theta(n)\) notation, as the sum increases directly with the number of terms.

• Purpose: To determine efficiency and scalability of algorithms.
• Focus: Primarily on the order of growth rather than exact values.

Arithmetic Sum

An arithmetic sum arises in sequences where each term is equal to the preceding term plus a constant difference. In our analyzed series with a common ratio of \(r=1\), each term is \(1\), making it a simple arithmetic progression.
For example, the sequence \(1 + 1 + 1 + \cdots + 1\) is straightforward in structure. Here, every term remains constant, so the sum is simply the constant multiplied by the number of terms, i.e., if there are \(n\) terms, then the sum is \(n\) times the constant. In our case, when the number of terms is \(n+1\), the sum equals \(n+1\).

• Example: With \(n=3\), the series \(1 + 1 + 1 + 1\) has 4 terms, summing to 4.
• Application: Common in problems dealing with equal value distribution over a set of iterations.