Sums of Cubes

1
Sums of Cubes

• Finding a Summation Formula
• Sum as a Polynomial
• Properties of the Summation Polynomial
• Solving the Coefficients
• Summation Formulae
• Proof by Induction

2
Finding a Summation Formula
Problem
3
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.
4
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.
5
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.
Solution
6
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.
Here we have only reversed the order of summation.
This has important applications in integration.
Solution
7
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.
Here we have only reversed the order of summation.
This has important applications in integration.
Solution
8
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.
Here we have only reversed the order of summation.
This has important applications in integration.
Solution
9
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.
Here we have only reversed the order of summation.
This has important applications in integration.
Solution
Conclude
10
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.
Here we have only reversed the order of summation.
This has important applications in integration.
Solution
Conclude
11
Sum as a Polynomial
Problem
12
Sum as a Polynomial
Problem
Conclusion
13
Sum as a Polynomial
Problem
Conclusion

• Properties of the polynomial S(n)
• S(0) 0.
• S(n 1) S(n) (n 1)3.

14
Sum as a Polynomial
Problem
Conclusion

• Properties of the polynomial S(n)
• S(0) 0.
• S(n 1) S(n) (n 1)3.

Here we have only used the definition of the sum
S(n).
15
Sum as a Polynomial
Problem
Conclusion

• Properties of the polynomial S(n)
• S(0) 0.
• S(n 1) S(n) (n 1)3.

Here we have only used the definition of the sum
S(n).
16
Sum as a Polynomial
Problem
Conclusion

• Properties of the polynomial S(n)
• S(0) 0.
• S(n 1) S(n) (n 1)3.

Here we have only used the definition of the sum
S(n).
Method
Solve the coefficients ak using the conditions
1 and 2 for the polynomial S(n).
17
Properties of the Summation Polynomial
Conditions
18
Properties of the Summation Polynomial
Conditions
19
Properties of the Summation Polynomial
Conditions
20
Properties of the Summation Polynomial
Conditions
To determine the coefficients ak, k 1,,3, we
use the condition 2.
21
Properties of the Summation Polynomial
Conditions
To determine the coefficients ak, k 1,,4, we
use the condition 2.
22
Properties of the Summation Polynomial
Conditions
To determine the coefficients ak, k 1,,3, we
use the condition 2.
23
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.
24
Solving the Coefficients
Conditions
25
Solving the Coefficients
Conditions
26
Solving the Coefficients
Conditions
27
Solving the Coefficients
Conditions
We solve this system of linear equations by
elimination. The 2nd equation yields a41/4.
Substituting that to the 3rd equation yields
a31/2. Substituting these values to the 4th
equation yields a21/4. The value of a1 follows
then from the last equation.
28
Solving the Coefficients
Conditions
We solve this system of linear equations by
elimination. The 2nd equation yields a41/4.
Substituting that to the 3rd equation yields
a31/2. Substituting these values to the 4th
equation yields a21/4. The value of a1 follows
then from the last equation.
29
Summation Formulae
Formulae
The previous 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
30
Proof by Induction
Formula
The previous considerations already show that the
above formula is correct.
31
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.
32
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
00. This is clearly true.
33
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
00. This is clearly true.
2
34
Proof by Induction
Formula
Proof by Induction (contd)
2
35
Proof by Induction
Formula
Proof by Induction (contd)
2
3
36
Proof by Induction
Formula
Proof by Induction (contd)
2
3
37
Proof by Induction
Formula
Proof by Induction (contd)
2
3
38
Proof by Induction
Formula
Proof by Induction (contd)
2
3
39
Proof by Induction
Formula
Proof by Induction (contd)
2
3