Problem 34 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{5} i^{i-1}$$

Step-by-Step Solution

Answer

The sum of the series is 701.

Step 1: Identify the terms of the series

The given series is \( \sum_{i=1}^{5} i^{i-1} \). This means we need to sum the values of \( i^{i-1} \) for each integer \( i \) from 1 to 5.

Step 2: Calculate each term individually

Let's evaluate each term separately: 1. For \( i=1 \), the term is \( 1^{1-1} = 1^0 = 1 \).2. For \( i=2 \), the term is \( 2^{2-1} = 2^1 = 2 \).3. For \( i=3 \), the term is \( 3^{3-1} = 3^2 = 9 \).4. For \( i=4 \), the term is \( 4^{4-1} = 4^3 = 64 \).5. For \( i=5 \), the term is \( 5^{5-1} = 5^4 = 625 \).

Step 3: Sum the calculated terms

Add all the individual terms calculated in Step 2:\[1 + 2 + 9 + 64 + 625 = 701\]

Step 4: Conclude the solution

The sum of the series \( \sum_{i=1}^{5} i^{i-1} \) is 701.

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \(f(x) = a^x\), where \(a\) is a constant known as the base, and \(x\) is the exponent. The key feature of exponential functions is that the variable \(x\) appears as the exponent, which results in the function growing very quickly or decaying very rapidly, depending on the value of \(a\).
Exponential functions are widely used in real-world applications such as population growth models, radioactive decay, and interest calculations in finance. In our original exercise, each term \(i^{i-1}\) is an example of an exponential expression where both the base and the exponent come from the sequence of natural numbers starting from \(i = 1\).
Understanding how to compute these expressions is crucial for dealing with series that involve terms with varying exponents. By calculating expressions like \(i^{i-1}\), we practice evaluating exponential terms step-by-step, which is a fundamental skill in mathematics.

Summation Notation

Summation notation, denoted by the Greek letter \(\Sigma\), is a shorthand method used to represent the sum of a sequence of numbers. When expressed as \(\sum_{i=1}^{n} a_i\), it indicates that you should sum all terms from the index value \(i = 1\) to \(i = n\).
This notation is incredibly helpful as it eliminates the need to write long expressions manually. Instead of listing all terms, you can compactly express the idea of a series or repeated addition. For example, our original exercise uses summation notation \(\sum_{i=1}^{5} i^{i-1}\), meaning we calculate each \(i^{i-1}\) from \(i = 1\) to 5 and then add the results.
The conventions of summation notation include using letters like \(i, j, k\) as index variables, and the expressions following the \(\Sigma\) symbol can be any function or expression involving the index variable. This feature makes summation notation a powerful tool for mathematicians when working with series and sequences.

Mathematical Series

Mathematical series involve the sum of terms from a sequence, and they play a critical role in different branches of mathematics. A series can be finite, like in our original step by step solution with limits from 1 to 5, or infinite, involving an endless sequence of terms.
Series are used to approximate real numbers and functions, solve equations, and analyze mathematical models. In our original exercise, we deal with a finite series, represented as \(\sum_{i=1}^{5} i^{i-1}\), which involves evaluating exponential expressions \(i^{i-1}\) for each integer \(i\) from 1 to 5, and summing them all up to get a total.
By understanding series, students learn to manipulate expressions, simplify computations, and identify patterns, all of which are foundational skills in calculus and higher mathematics. Mathematicians often look at both arithmetic and geometric series, providing insight into different types of growth and decay.