Problem 44 - College Algebra

Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$1^{4}+2^{4}+3^{4}+\dots+12^{4}$$

Step-by-Step Solution

Answer

The given series can be expressed in summation notation as \(\sum_{i=1}^{12} i^4\).

Step 1: Identify the General Term

In this problem, each term in the series is the fourth power of a natural number. The general term for this sequence can thus be written as \(i^{4}\).

Step 2: Identify the Lower and Upper Limit

The lower limit for the series is given. It is 1 because the series starts at \(1^4\). The upper limit is the number \(n\) that satisfies \(n^4\) as the last term in the series. The series stops at \(12^4\), so \(n = 12\).

Step 3: Write in Summation Notation

Put everything together into summation (sigma) notation. The general term is \(i^4\), the lower limit is 1, and the upper limit is 12. The series can therefore be expressed in summation notation as \(\sum_{i=1}^{12} i^4\).

Key Concepts

General Term of a Series

When we talk about the general term of a series, we're referring to a formula that represents the terms of the sequence. This concept is central to understanding sequences and series, as it allows us to express and work with all the terms in a uniform way. In the given exercise, each term is a power of a natural number, specifically the fourth power. Hence, the general term for the series is written as \(i^4\), where \(i\) represents the position of the term in the sequence.

For instance, if we're looking at the first term, we substitute \(i=1\) to get \(1^4\), for the second term, we substitute \(i=2\) to get \(2^4\), and this pattern continues systematically up to the sequence's end. This method helps in identifying the pattern of the series, which is crucial in writing the series in a compact form, notably the summation notation.

Upper and Lower Limits

In summation notation, specifying the upper and lower limits is crucial; they determine where the series begins and ends. The lower limit is displayed directly below the summation symbol (sigma), while the upper limit is shown above it. In this exercise, the given lower limit is 1, suggesting that the index of summation starts at the natural number 1. The upper limit in the problem is 12, which is the number such that \(12^4\) is the last term in the series.

• The lower limit, 1, corresponds to \(1^4\), the first term in the series.
• The upper limit, 12, corresponds to \(12^4\), the last term in the series.

Thus, the limits dictate that we are considering all natural numbers from 1 to 12, raising each to the fourth power in order to form our series. These limits aid in the precise and concise expression of lengthy series. Establishing the correct limits is essential to ensure accuracy in calculations and representations.

Sigma Notation

The concept of sigma notation serves as a compact way to represent series. It is identified by the Greek letter sigma (\(\Sigma\)). This notation simplifies the representation of a series by indicating the general term and the range of values the index can take—marked by the upper and lower limits. In the context of our textbook exercise, sigma notation is used to express the given series compactly.

To construct this notation, three components are necessary:

• The general term, which in this case is \(i^4\), designates the pattern of terms in the series.
• The lower limit of summation, which is 1, indicates the starting point of the series.
• The upper limit of summation, which is 12, tells us the series ends at the term \(12^4\).

Put together, the series \(1^4 + 2^4 + 3^4 + \dots + 12^4\) is succinctly written in sigma notation as:\[ \sum_{i=1}^{12} i^4 \]. Understanding and applying sigma notation helps students and mathematicians alike to work efficiently with series and perform various operations such as finding the sum or average.