Problem 33 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{4} i^{i}$$

Step-by-Step Solution

Answer

The sum is 288.

Step 1: Understand the Series

The series is given by \( \sum_{i=1}^{4} i^{i} \). This requires us to find the sum of each term \( i^{i} \) from \( i = 1 \) to \( i = 4 \).

Step 2: Calculate Individual Terms on Raising to Power

Compute each term by raising \( i \) to the power of \( i \): - When \( i = 1 \), \( 1^{1} = 1 \).- When \( i = 2 \), \( 2^{2} = 4 \).- When \( i = 3 \), \( 3^{3} = 27 \).- When \( i = 4 \), \( 4^{4} = 256 \).

Step 3: Sum the Individual Terms

Add all the computed terms together. - So, the sum is \( 1 + 4 + 27 + 256 \).

Step 4: Compute the Final Sum

Perform the addition: \( 1 + 4 + 27 + 256 = 288 \).

Key Concepts

Understanding Power Functions

In mathematics, a power function refers to a function of the form \( f(x) = x^{n} \), where \( n \) is a real number and \( x \) is a variable. This means we are multiplying \( x \) by itself \( n \) times. For example, in the series \( \sum_{i=1}^{4} i^{i} \), each term like \( i^{i} \) is formed by raising the number \( i \) to its own power.
This concept is essential in identifying the pattern or rule that dictates the behavior of the sequence's terms.
Power functions often have distinctive characteristics depending on the value of \( n \):

• If \( n \) is a positive integer, generally the function increases as \( x \) increases.
• If \( n \) is zero, the power function becomes a constant (as \( x^{0} = 1 \) for any \( x \) except zero).
• If \( n \) is negative, the function decreases as \( x \) increases.

Understanding how power functions work allows us to efficiently compute each term in a series that involves exponentiation.

The Process of Series Calculation

Series calculation is all about summing up terms that follow a specific pattern or rule. In our example, the pattern is given by \( i^{i} \). Calculating a series involves several steps:

• Identify the pattern: Here, each term is raised to the power of itself.
• Compute each term: You determine the value of each term individually. For instance, when \( i = 3 \), you calculate \( 3^{3} = 27 \).


• Add the terms: Once you've calculated the individual terms, sum them up to get the total of the series. This is straightforward, simply addition of the identified terms.

By following these steps, as in our example, you calculate \( 1 + 4 + 27 + 256 = 288 \) to find the total sum. This systematic approach ensures accuracy in your calculations.

What is a Mathematical Series?

A mathematical series is essentially the summation of a sequence of terms. This sequence of numbers is generated from some function or rule. In our example, it’s specifically a finite series which means it has a limited number of terms, from 1 to 4.
Even though our series \( \sum_{i=1}^{4} i^{i} \) is finite, series in mathematics can be finite or infinite:

• Finite Series: A series that stops after a certain number of terms. In our case, it ends at \( i = 4 \).
• Infinite Series: Continues indefinitely. For example, the sum of an infinite geometric sequence.

Mathematical series can often be found in daily life applications such as calculations of interests, analysing waves in physics, and many more. Understanding mathematical series and how to calculate them can help one tackle complex numerical and real-world problems efficiently.