Problem 17 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write each series in sigma notation. $$ 1^{1}+2^{2}+3^{3}+4^{4}+5^{5} $$

Step-by-Step Solution

Answer

The series in sigma notation is \( \sum_{n=1}^{5} n^n \).

Step 1: Identify the Pattern

In the given series, the terms are in the form of \( n^n \). The exponents and bases are the same, increasing by 1 with each term. This means the pattern is of the form \( n^n \) starting from \( n=1 \) up to the maximum term.

Step 2: Determine the Series' Range

Observe that the series starts from \( 1^1 \) and goes to \( 5^5 \). Therefore, the range of the series is from \( n=1 \) to \( n=5 \).

Step 3: Express in Sigma Notation

Using the pattern \( n^n \) found in Step 1, and the range of terms from Step 2, we can express the series using sigma notation as \( \sum_{n=1}^{5} n^n \). This means that you sum the terms \( n^n \) from \( n = 1 \) to \( n = 5 \).

Key Concepts

Series Representation

When we talk about a series in mathematics, we are discussing the sum of the terms of a sequence. Putting a series into sigma notation is a powerful way to represent repeated addition concisely. Consider our example from the exercise: the series given is \(1^1 + 2^2 + 3^3 + 4^4 + 5^5\). Each term follows a specific pattern,
which when summed, forms a series.Sigma notation uses the Greek letter \(\Sigma\) to denote this sum. It provides a way to see more easily and clearly the starting point, ending point, and the pattern in the summation of a series. By stating the series as \(\sum_{n=1}^{5} n^n\), it communicates that we start at \(n = 1\) and end at \(n = 5\), with each term being of the form \(n^n\).
Representing series in this way is concise and helps simplify complex mathematical expressions.

Pattern Recognition

Pattern recognition is a key skill when working with series, especially when you need to express them in sigma notation. Recognizing how terms are structured allows you to identify the underlying formula that describes each term in the sequence.
In our exercise, the pattern is sensed by noticing each term is raised to a power that matches its base: \(1^1, 2^2, 3^3, 4^4, 5^5\).Understanding the role of each part is essential:

• Base: The base is the number itself, increasing linearly from 1 to 5.


• Exponent: The exponent is the same as the base in this series, and it also simply increases.

By seeing this pattern, it allows you to express it in a clean formula such as \(n^n\), where \(n\) is an integer starting at 1 and moving to 5. Recognizing patterns in such a structured way can greatly reduce complexity in mathematics.

Exponential Expressions

An exponential expression is one where a base is raised to a power or exponent. In mathematics, exponentiation is a critical operation which shows how many times a number, the base, is multiplied by itself. In our specific series example, we encounter terms like \(n^n\), where both the base and exponent are the same.Some properties of exponential expressions include:

• If the exponent is 1, the expression simply evaluates to the base itself.


• As the exponent increases, the value of the expression can grow very rapidly.


• Exponential growth is notably different from linear growth for this reason.

In our series of \(1^1 + 2^2 + 3^3 + 4^4 + 5^5\), this type of growth is evident, making the series a good example of exponential operations where each step in the series grows increasingly larger due to both the base and exponent increasing simultaneously.