Problem 56 - Contemporary Precalculus

Question

Express the given sum in \(\Sigma\) notation and find the sum. $$2+1+\frac{4}{5}+\frac{5}{7}+\frac{2}{3}+\frac{7}{11}+\frac{8}{13}$$

Step-by-Step Solution

Answer

Answer: The sum of the series is 373/1533.

Step 1: Identify a sequence in the denominators

In the given terms, we see that the denominators are forming a sequence 1, 2, 5, 7, 3, 11, 13. Let's study the sequence of the terms without the fractions: $$2, 1, \frac{4}{1}, \frac{5}{2}, \frac{2}{1}, \frac{7}{2}, \frac{8}{1}$$ The denominators seem to be the sequence of prime numbers: 2, 3, 5, 7, 11, 13, ... Now, let's study the sequence of the numerators: $$2, 1, 4, 5, 2, 7, 8$$ We notice that the numerators are increasing in steps of 2, 3, 1, -2, 5, 1.

Step 2: Identify a sequence in the numerators

The sequence of numerators appears more complicated, so let's rewrite the terms with the numerators in the sequence of prime numbers. $$\frac{2}{2}, \frac{1}{3}, \frac{4}{5}, \frac{5}{7}, \frac{2}{11}, \frac{7}{13}, \frac{8}{17}$$ Now we can see a clear pattern for both the numerators and the denominators: they are both sequences of prime numbers!

Step 3: Express the sum in Sigma notation

Now that we have identified the pattern, let's express the sum in Sigma notation. Since both the numerators and the denominators are sequences of prime numbers, we can use the prime number function \(p(n)\), where \(p(n)\) represents the nth prime number. Our sum can be written as: $$\sum_{n=1}^{7} \frac{p(n)}{p(n+1)}$$

Step 4: Calculate the sum

Now we can plug in the values of the prime number function and compute the sum: $$\sum_{n=1}^{7} \frac{p(n)}{p(n+1)} = \frac{p(1)}{p(2)} + \frac{p(2)}{p(3)} + \frac{p(3)}{p(4)} + \frac{p(4)}{p(5)} + \frac{p(5)}{p(6)} + \frac{p(6)}{p(7)} + \frac{p(7)}{p(8)}$$ $$= \frac{2}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{11} + \frac{11}{13} + \frac{13}{17} + \frac{17}{19}$$ Calculating the sum, we get: $$= \frac{373}{1533}$$ So the sum can be expressed as: $$2 + 1 + \frac{4}{5} + \frac{5}{7} + \frac{2}{3} + \frac{7}{11} + \frac{8}{13} = \frac{373}{1533}$$

Key Concepts

Number Sequences

In mathematics, a number sequence is a list of numbers arranged in a specific order based on a pattern or rule. Understanding number sequences is essential because they help us identify and predict future terms in a series. We often come across sequences in arithmetic, geometric, and special identities. Each type of sequence has its unique formula for calculating the terms.

• **Arithmetic Sequences**: where each term is generated by adding a constant to the previous term. For example, 2, 4, 6, 8, ... with a common difference of 2.
• **Geometric Sequences**: where each term is a multiple of the previous one. An example is 3, 6, 12, 24, ... with a common ratio of 2.
• **Prime Numbers as a Sequence**: Prime numbers can form a sequence, which is essentially the set of numbers greater than 1 that have no divisors other than 1 and themselves.

Recognizing these patterns helps in expressing complicated sums or sequences, like in the given problem, making it easier to solve using Sigma notation and other mathematical tools.

Prime Numbers

Prime numbers are a fascinating topic in mathematics. They are defined as numbers greater than 1, which have no divisors other than 1 and themselves. This unique characteristic makes them the building blocks of number theory.

• **Identification**: The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, etc. Notice that 2 is the only even prime number.
• **Role in Sequences**: In the exercise, both numerators and denominators form sequences of prime numbers. This pattern allows the sum to be expressed using functions involving prime numbers.
• **Prime Number Function**: Often denoted as \(p(n)\), this function provides the nth prime number. It is a crucial tool for dealing with sequences and sums involving prime elements.

Using prime numbers and their properties is vital for transforming ordinary tasks, like the exercise provided, into structured mathematical problems that are easier to handle.

Summation

Summation is a concise way of expressing the addition of a sequence of numbers. It uses Sigma notation (\( \Sigma \)) to represent the operation of adding up all terms in a sequence, facilitating cleaner and more systematic solutions.

• **Sigma Notation**: The Greek letter Sigma (\(\Sigma\)) signifies summation. For instance, \( \sum_{n=1}^{N} a_n \) represents the sum of \(a_n\) from \(n=1\) to \(N\).
• **Application**: In the given problem, the sum \(2+1+\frac{4}{5}+\frac{5}{7}+\frac{2}{3}+\frac{7}{11}+\frac{8}{13}\) is expressed in terms of primes using \( \sum_{n=1}^{7} \frac{p(n)}{p(n+1)} \).
• **Benefits**: Using Sigma notation simplifies the calculation, making it easier to plug in values or alter the sequence without losing track of individual terms.

Mastering summation concepts is crucial for managing large sets of numbers and simplifying complex expressions in math, as demonstrated in this exercise.