Problem 104 - Precalculus with Limits

Question

In Exercises 103-112, use sigma notation to write the sum. \( \dfrac{5}{1 + 1} + \dfrac{5}{1 + 2} + \dfrac{5}{1 + 3} + \cdots + \dfrac{5}{1 + 15} \)

Step-by-Step Solution

Answer

The sum \( \dfrac{5}{1 + 1} + \dfrac{5}{1 + 2} + \dfrac{5}{1 + 3} + \cdots + \dfrac{5}{1 + 15} \) written in the sigma notation is: \( \sum_{n=2}^{16} \frac{5}{n} \).

Step 1: Identify Common Factor

The common factor is the part of the term that does not change as we move from one term to the next. In this case, that is 5, which is the numerator in every fraction.

Step 2: Identify the Variable

The denominator in each term changes as we move from one term to the next, with every new term's denominator one integer higher than the previous one. It starts from 1+1=2 in the first term and increases to 1+15=16 in the last term.

Step 3: Write in Sigma Notation

Now that we've identified the common factor and the variable pattern, we can express the series using sigma notation. The sum can be represented as: \( \sum_{n=2}^{16} \frac{5}{n} \).

Key Concepts

Series and Sequences

When we talk about series and sequences, we're looking at a list of numbers following a particular pattern. A sequence is essentially a list of items, usually numbers, that are arranged in a specific order. For example, the sequence of numbers that starts with 1 and increases by 1 each time, like this: 1, 2, 3, 4, ..., is a simple instance of a sequence. A sequence can be finite or infinite, depending on whether it has an end or goes on forever.

A series, on the other hand, is what you get when you add up all the terms in a sequence. In other words, it's the sum of the sequence elements. In the exercise we're looking at, the series is achieved by adding up a sequence of terms where each term consists of 5 divided by an increasing integer, starting with 2 up until 16. This kind of series requires an understanding of summation notation to both write and understand it effectively.

Summation Notation

The summation notation is a mathematical shorthand that allows us to express the addition of a series of terms without having to write them all out. This is done using the Greek letter sigma (\f\(\f\)), which stands for 'sum'.

The general form of summation notation is: \f\(\f\underline{\phantom{xxx}}\f\) where \f\(i\f\) is the index of summation, \f\(a_i\f\) represents the terms in the series, \f\(m\f\) is the lower bound of summation, and \f\(n\f\) is the upper bound of summation. The concept is to sum up all terms \f\(a_i\f\) starting with the term where \f\(i\f\) equals the lower bound \f\(m\f\) and ending with the term where \f\(i\f\) equals the upper bound \f\(n\f\).

In reference to the textbook exercise, the series \f\(\f\underline{\phantom{xxx}}\f\)dfrac{5}{1 + 1} + \f\(dfrac{5}{1 + 2} + \f\)dfrac{5}{1 + 3} + \f\(cdots + \f\)dfrac{5}{1 + 15} \f\( can be concisely written using summation notation as \f\)\f\underline{\phantom{xxx}}\f\(sum_{n=2}^{16} \f\)frac{5}{n} \f$. This neatly captures the entire series without writing out each individual fraction.

Arithmetic Sequences

In an arithmetic sequence, each term is created by adding a constant value, known as the common difference, to the previous term. This means that the difference between consecutive terms is always the same. For example, if we have a sequence that starts with 1 and increases by 3 each time, the sequence would go 1, 4, 7, 10, ..., and so forth. Each term after the first is the sum of the common difference (3, in this case) and the preceding term. This type of sequence is common in mathematical problems because of its regularity.

In our example from the textbook exercise, although the terms themselves do not form an arithmetic sequence, the denominators do. The denominator begins at 2 (1+1) and increases by 1 each time, forming the arithmetic sequence 2, 3, 4, ..., 16. The common difference here is 1. Recognizing this pattern allows us to understand the structure of the series and use the properties of arithmetic sequences to analyze or simplify the expression.