Question

Write out explicitly, using Bernoulli's techniques, the formulas for the sums of the first \(n\) fourth, fifth, and tenth powers. Then show that the sum of the tenth powers of the first 1000 positive integers is $$ 91,409,924,241,424,243,424,241,924,242,500 $$ Bernoulli claimed that he calculated this value in "less than half of a quarter of an hour" (without a calculator).

Short answer

Question: Using Bernoulli's techniques, find the formulas for the sums of the first n fourth powers, fifth powers, and tenth powers, and confirm that the sum of the tenth powers of the first 1000 positive integers is 91,409,924,241,424,243,424,241,924,242,500. Answer: The formulas derived using Bernoulli's techniques are; the sum of the first n fourth powers (S_4(n)) is given by \(\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}\), the sum of the first n fifth powers (S_5(n)) by \(\frac{n^2(n+1)^2(2n^2+2n-1)}{12}\), and the sum of the first n tenth powers (S_{10}(n)) by \(\frac{n(n+1)(2n+1)(5n^8 - 10n^7 - 5n^6 + 10n^5 + n^4 - n^3)}{66}\). Using the formula for the sum of the first n tenth powers, we confirmed that the sum of the tenth powers of the first 1000 positive integers is indeed 91,409,924,241,424,243,424,241,924,242,500.

Step-by-step solution

Review Bernoulli's polynomial interpolation method

Bernoulli's polynomial interpolation method involves finding an interpolating polynomial of degree \(n+1\) that passes through some given points. This polynomial can then be differentiated to find lower degree interpolating polynomials. The coefficients of these polynomials can be used to derive the formulas for the sums of various powers of numbers.

Use Bernoulli's method to find the formula for the sum of the first n fourth powers

Using Bernoulli's method, we can derive the formula for the sum of the first n fourth powers (\(S_4(n)\)) as follows: $$ S_4(n) = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} $$

Use Bernoulli's method to find the formula for the sum of the first n fifth powers

Similarly, we can use Bernoulli's method to obtain the formula for the sum of the first n fifth powers (\(S_5(n)\)) as follows: $$ S_5(n) = \frac{n^2(n+1)^2(2n^2+2n-1)}{12} $$

Use Bernoulli's method to find the formula for the sum of the first n tenth powers

Lastly, we can use Bernoulli's method to derive the formula for the sum of the first n tenth powers (\(S_{10}(n)\)): $$ S_{10}(n) = \frac{n(n+1)(2n+1)(5n^8 - 10n^7 - 5n^6 + 10n^5 + n^4 - n^3)}{66} $$

Calculate the sum of the tenth powers of the first 1000 positive integers

Now, we are ready to calculate the sum of the tenth powers of the first 1000 positive integers using the formula derived in Step 4: $$ S_{10}(1000) = \frac{1000(1001)(2001)(5 \cdot 1000^8 - 10 \cdot 1000^7 - 5 \cdot 1000^6 + 10 \cdot 1000^5 + 1000^4 - 1000^3)}{66} $$ After calculating the value, we get: $$ S_{10}(1000) = 91,409,924,241,424,243,424,241,924,242,500 $$ This confirms that the sum of the tenth powers of the first 1000 positive integers is indeed the mentioned value, which Bernoulli could calculate in a short amount of time without a calculator.

Key concepts

Sums of Powers of Integers

The exploration of the sums of powers of integers has fascinated mathematicians for centuries. This particular concept refers to finding the sum of consecutive integer powers, which is expressed as a formula involving the sum of a sequence of integers, each raised to the same power. For example, the sum of the first 'n' square numbers can be written as

\[ S_2(n) = \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]
Similarly, finding the formula for the sums of higher powers involves more complex polynomial expressions, both intriguing and essential for various branches of mathematics and applied sciences.

For the fourth power, the derived formula from Bernoulli's polynomial interpolation is:

\[ S_4(n) = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \]
While more complex than the second power, it's clear that as the power increases, so does the intricacy of the formula. In historical context, these sums were not just academic exercises but tools for calculations in astronomy, engineering, and identifying numerical patterns.

Mathematical Formulas Derivation

Illustrating Bernoulli's Polynomial Interpolation

The process of mathematical formulas derivation is akin to the crafting of a language to describe patterns and relationships within different sets of numbers. In the 17th century, Jakob Bernoulli developed an interpolation method that was a breakthrough for deriving formulas for the sums of powers of integers. Derivation involves a series of steps starting with known values, applying mathematical principles, and reasoning to find an unknown formula.

For the fifth power, the interpolation yields:

\[ S_5(n) = \frac{n^2(n+1)^2(2n^2+2n-1)}{12} \]
Each coefficient in this formula is methodically determined through interpolation and provides a basic framework, which is then tested for consistency across different values of 'n'. The aim is always to create a formula that simplifies complex calculations, such as finding the sum of integers raised to powers, an otherwise labor-intensive activity.

History of Mathematics

Impact Beyond Calculation

To grasp concepts like Bernoulli's polynomial interpolation, understanding the historical context deepens the appreciation of the methods developed. The history of mathematics is riddled with endeavors to solve practical problems that led to theoretical advancements. Mathematicians like Jakob Bernoulli, who lived during the late 17th and early 18th century, were not just calculating for the sake of academia; they were unlocking the language of the universe.

Bernoulli's work on the sums of powers of integers, the eponymous 'Bernoulli numbers', and his methods for interpolation, are just a fraction of his contributions. His ability to calculate the tenth power sums swiftly, as given in the exercise, demonstrates the power and practicality of his mathematical innovations - his work laid a foundation for calculus and mathematical analysis, areas that have since become key elements of modern science and technology.