Sums of Cubes Finding a Summation Formula Sum

Sums of Cubes Finding a Summation Formula Sum as a Polynomial Properties of the Summation Polynomial Solving the Coefficients Summation Formulae Proof by Induction Index FAQ

Finding a Summation Formula Problem We will here derive the formula using methods which can be applied to compute sums of positive integer powers of integers from 0 to n. This has important applications in integration. Here we have only reversed the order of summation. This follows since S(n) is a sum of n polynomials each of degree 3 in n. Cancellation may happen, so the degree may be < 4. Solution Conclude Index Mika Seppälä: Sums of Cubes FAQ

Sum as a Polynomial Problem Conclusion Properties of the polynomial S(n): 1. S(0) = 0. 2. S(n + 1) = S(n) + (n + 1)3. Method Index Here we have only used the definition of the sum S(n). Solve the coefficients ak using the conditions 1 and 2 for the polynomial S(n). Mika Seppälä: Sums of Cubes FAQ

Properties of the Summation Polynomial Conditions To determine the coefficients ak, k = 1, …, 3, we use the condition 2. The last equation must hold for all values of n. The two polynomials on the different sides of the equation are the same if and only if the coefficients of the various order terms are the same. This gives equations for the coefficients ak. Index Mika Seppälä: Sums of Cubes FAQ

Solving the Coefficients Conditions We solve this system of linear equations by elimination. The 2 nd equation yields a 4=1/4. Substituting that to the 3 rd equation yields a 3=1/2. Substituting these values to the 4 th equation yields a 2=1/4. The value of a 1 follows then from the last equation. Index Mika Seppälä: Sums of Cubes FAQ

Summation The previous. Formulae arguments justified the formula 3 in the list below. Formulae for all sums of positive integer powers of integers can be found in this way. Here are some of them: 1 2 3 4 5 Index Mika Seppälä: Sums of Cubes FAQ

Proof by Induction Formula The previous considerations already show that the above formula is correct. As an example of Mathematical Induction we will, never the less, prove the above result. Proof by Induction 1 Inserting n = 0 in the above formula one gets 0=0. This is clearly true. 2 Index Mika Seppälä: Sums of Cubes FAQ

Proof by Induction Formula Proof by Induction (cont’d) 2 3 Here we apply the Induction Assumption to this sum. Index Expanding the right hand sides we see that these expressions are the same. Mika Seppälä: Sums of Cubes FAQ