Proof of the sum of the series ∑r³
Proof of the sum of the series ∑r³
Sum of a Series of Cubes
- A series in mathematics is the sum of a sequence of terms.
- The sum of the cubes of the terms of an arithmetic series is represented by the notation ∑r³ (sum of r cubed), where ‘r’ is the term number.
- A proven formula for the sum of cubes in an arithmetic series is ∑r³ = (n(n + 1)/2)², where ‘n’ is the number of terms.
Understanding the Proof
- The formula can be proven using induction, a mathematical proof technique.
- The base case is when n = 1, and indeed 1³ = (1(1 + 1)/2)².
- For the inductive step, assume the formula holds for n = k (that is, ∑r³ from r = 1 to k equals (k(k + 1)/2)²).
- Then we need to show that the formula holds for n = k + 1.
Applying the Inductive Hypothesis
- Adding (k + 1)³ to both sides of the inductive hypothesis results in ∑r³ from r = 1 to k + 1 equals (k(k + 1)/2)² + (k + 1)³ on the left side.
- The right side can be simplified to ((k + 1)(k + 2)/2)² by completing the square and simplifying.
- When both sides match, the formula is proven for n = k + 1 if assumed to be true for n = k. This concludes the proof by principle of mathematical induction.
Sum of Cubes in Practical Applications
- The formula for the sum of cubes can be applied in various mathematical problems, particularly in algorithms involving numerical computations and in statistical analysis.
- It also provides an easy reference for summing the cubes of consecutive integers or a specific range of integers.
Key Takeaway
- The important concept to remember here is the formula ∑r³ = (n(n + 1)/2)² and the approach of induction used to prove it. This is an essential aspect of understanding series and sequences in mathematics.