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Chapter 10: Problem 57

Evaluate each finite series. $$\sum_{n=0}^{4} 1^{n}$$

Short Answer

Expert verified
The sum of the series is 5.

Step by step solution

01

Understanding the Series

The given series is \( \sum_{n=0}^{4} 1^{n} \). This means we need to find the sum of the sequence of terms generated by raising 1 to the power of \( n \) for each \( n \) from 0 to 4.
02

Expanding the Series

Since \( 1^{n} = 1 \) for any integer \( n \), we expand the series \( \sum_{n=0}^{4} 1^{n} \) as follows: \( 1^{0} + 1^{1} + 1^{2} + 1^{3} + 1^{4} \). Since each \( 1^{n} \) is equal to 1, the expanded form becomes \( 1 + 1 + 1 + 1 + 1 \).
03

Calculating the Sum

Now, we sum the expanded terms: \( 1 + 1 + 1 + 1 + 1 = 5 \). This is the sum of the series.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

power of a number
The concept of the "power of a number" involves raising a number to an exponent. This means multiplying that number by itself a certain number of times. For example:
  • If you have a base of 2 and an exponent of 3, it means you multiply 2 by itself 3 times: \(2^3 = 2 \times 2 \times 2 = 8\).
  • The base number is the number you repeatedly multiply, while the exponent tells you how many times to multiply it by itself.
In the original exercise, each term from the series is formed by raising 1 to various powers, specifically from \(n=0\) to \(n=4\). Raising any number to the power of zero is always 1, and raising 1 to any power always results in 1, because there is no growth when multiplying 1 by itself multiple times. Understanding these rules helps in evaluating series such as \( \sum_{n=0}^{4} 1^{n} \).
sequence of terms
A "sequence of terms" in mathematics is simply an ordered list of numbers. Each number in the list is called a term. These terms are typically generated using a specific rule or formula based on their position in the sequence. For example:
  • A simple sequence could be the list of even numbers: 2, 4, 6, 8, ..., where each term can be determined by the formula \(2n\).
  • Another sequence might be the list of powers of 2: 1, 2, 4, 8, ..., where each term is \(2^n\).
In the exercise provided, the sequence of terms is generated by raising the number 1 to the power \(n\) for \(n=0\) to \(n=4\). The sequence becomes: 1, 1, 1, 1, 1. Each term here is 1 because 1 raised to any power is still 1. Recognizing sequences and how they are structured is crucial to understanding series and subsequent calculations.
sum of a series
The "sum of a series" refers to the result obtained when you add together all the terms of a sequence. A finite series, like the one in the exercise, has a specific beginning and end, meaning it consists of a limited number of terms. To compute the sum, simply add each term in the sequence sequentially. Here's how that works:
  • Consider the example: \(1 + 2 + 3 + 4 + 5\). The sum here is 15.
  • For the exercise series: \(1^{0} + 1^{1} + 1^{2} + 1^{3} + 1^{4}\), since each term equals 1, the sum is \(1 + 1 + 1 + 1 + 1 = 5\).
Understanding the process of summing these terms requires recognizing the consistency of the sequence, especially when each term is identical. Summation is a fundamental operation in mathematics and is often represented with the sigma notation \(\sum\). Knowing how to expand and sum series is an essential skill in problem-solving.

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