Problem 8 - A Gentle Introduction to the Art of Mathematics

Question

Evaluate the sum $$ \sum_{i=1}^{10} \frac{1}{i} \cdot[i \text { is prime }] $$

Step-by-Step Solution

Answer

1.176

Step 1: Understand the summation

The expression \(\text{[i is prime]}\) is an indicator function that equals 1 if \(i\) is a prime number and 0 otherwise. Therefore, the sum can be simplified by only considering terms where \(i\) is prime.

Step 2: Identify prime numbers from 1 to 10

The prime numbers between 1 and 10 are 2, 3, 5, and 7.

Step 3: Substitute primes into the sum

Substitute the identified prime numbers back into the sum expression: \( \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} \).

Step 4: Calculate each term

Calculate each term separately: \( \frac{1}{2} = 0.5, \ \frac{1}{3} \approx 0.333, \ \frac{1}{5} = 0.2, \ \frac{1}{7} \approx 0.143 \).

Step 5: Add the terms

Sum the values obtained: \(0.5 + 0.333 + 0.2 + 0.143 = 1.176\).

Key Concepts

Prime Numbers

Prime numbers are the building blocks of mathematics. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number is only divisible by 1 and itself. Some well-known examples of prime numbers are 2, 3, 5, and 7.
Prime numbers are important in various fields such as cryptography, number theory, and computer science.
Identifying prime numbers within a range, like from 1 to 10, is straightforward: just test each number to see if it has any divisors other than 1 and itself. For instance, the primes between 1 and 10 are:

• 2 (divisors: 1, 2)
• 3 (divisors: 1, 3)
• 5 (divisors: 1, 5)
• 7 (divisors: 1, 7)

Prime numbers play a vital role in many mathematical problems and solutions.

Indicator Function

An indicator function, also known as a characteristic function, is used in mathematics to indicate the presence of a certain property within a set. It takes the value 1 if a certain condition holds and 0 otherwise. For example, in the given exercise, \( \text{[i is prime]} \) is an indicator function that yields 1 when the number \(i\) is a prime number and 0 when it's not.
This function helps in selectively considering specific elements of a summation. Rather than checking each element individually, the indicator function streamlines calculations by filtering out non-relevant elements. In our example, the function only 'activates' the terms where \(i\) is a prime number. With this, identifying and processing only the prime-numbered terms of a sequence becomes more efficient and systematic.

Summation

Summation, often denoted by the Greek letter sigma (\( \Sigma \)), is the addition of a sequence of numbers. In mathematical terms, \( \sum_{i=a}^{b} f(i) \) refers to summing up the values of the function \( f(i) \) from \( i=a \) to \( i=b \). Summation can involve simple sequences, like the sum of all integers from 1 to 10, or more complex ones like in our example.
In the given exercise, the summation is special because it includes a product of \( \frac{1}{i} \) and an indicator function. We simplify the process by filtering out non-prime \( i \) using the indicator function. This results in:

• Identifying prime numbers (2, 3, 5, 7)
• Calculating their reciprocals: \( \frac{1}{2}, \frac{1}{3}, \frac{1}{5}, \frac{1}{7} \)
• Adding these values: \( 0.5 + 0.333 + 0.2 + 0.143 = 1.176 \)

Summation is a fundamental concept that allows complex calculations to be broken down into smaller, manageable steps. Understanding summation and how to work with it is crucial for solving numerous mathematical problems.