Problem 8 - Essential Calculus

Question

In Exercises 7 -12, use sigma notation to write the sum. $$ \frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\cdots+\frac{5}{1+15} $$

Step-by-Step Solution

Answer

The sum expressed in sigma notation is \(\sum_{n=1}^{15} \frac{5}{1+n}\).

Step 1: Identify the summation pattern and the terms

The first step is to observe the pattern of the series given. We can immediately notice that the numerators are always 5 and the denominators gradually increase by 1 for each term in the series, starting from 2 (1+1) to 16 (1+15).

Step 2: Express the sum in sigma notation

Sigma notation is a concise way of expressing long summation series. We denote \(n\) as the variable that changes in each term, here \(n\) is ranging from 1 to 15 (because the series starts from \(1+1\) and ends in \(1+15\)). The general term of the series can be written as \(\frac{5}{1+n}\). So, the sum in sigma notation is \(\sum_{n=1}^{15} \frac{5}{1+n}\).

Key Concepts

Series and Sequences

In mathematics, both series and sequences are fundamental concepts used to understand lists of numbers and their behaviors. A sequence is essentially a list of numbers arranged in a specific order, where each number is called a term. Every sequence has a defined pattern or rule that determines how each term is found in relation to its position. For example, in the sequence 2, 4, 6, 8, each term increases by 2 from the previous one. This clear pattern defines it as an arithmetic sequence.

A series, on the other hand, is the sum of the terms of a sequence. When you add up all the elements of a sequence, you get a series. For example, if we sum the first four terms of the sequence mentioned above, we get the series 2 + 4 + 6 + 8, which equals 20. The difference is that while a sequence lists individual numbers, a series combines them through addition, giving a single sum.

Understanding the difference between a sequence and a series helps in writing, interpreting, and solving problems involving patterns of numbers in math.

Summation Notation

Summation notation, often called sigma notation, is a mathematical symbol that provides a compact way to express the sum of a series. The Greek letter sigma (\( \Sigma \)) represents the summation. When you see this symbol, it signals that you are adding up a list of numbers according to some rule.

In sigma notation, a series is expressed with the format \( \sum_{i=a}^{b} expression \). Here, \( i \) is the index of summation, representing each term in the series; \( a \) is the lower bound, showing where summation starts; and \( b \) is the upper bound, where the summation ends. The "expression" represents the general term of the series.

For example, the series \( \frac{5}{2}+\frac{5}{3}+\cdots+\frac{5}{16} \) can be neatly written as \( \sum_{n=1}^{15} \frac{5}{1+n} \). This notation succinctly encapsulates the entire sequence of terms to be summed by using the pattern of \( \frac{5}{1+n} \) as \( n \) varies from 1 to 15.

Mathematical Patterns

Mathematical patterns form the backbone of sequence and series problems. Patterns enable us to predict subsequent terms or efficiently perform calculations without manually adding each term one by one.

These patterns can be arithmetic, where each term is obtained by adding a constant to the previous term; geometric, where each term is found by multiplying the previous one by a constant; or they could follow more complex rules.

Identifying the pattern in a sequence, like in the given problem where the fraction's denominator increases by 1 each step, helps in formulating it in summation notation. To recognize these patterns, it's crucial to examine changes from term to term and determine the rule the sequence follows. This insight is what transforms the problem-solving process, allowing significant numbers of terms to be expressed through concise mathematical expressions like sigma notation.

So, recognizing and working with mathematical patterns simplifies computations and offers powerful tools for understanding complex sequences and series in mathematics.