March 23, 2018 Friday

Bedtime Story 

Faulhaber’s Formula

     

Tonight we shall in our bedtime story go totally mathematical and consider the Faulhaber’s formula which is sometimes also known as Bernoulli’s formula.

To many it may be distasteful but mathematics is so fundamental to our lives that no matter how it tastes and feels to our minds, it has to be dealt with and tackled up front mercilessly without giving in to our subjective feelings which are any way deeply and outrageously flawed.

Faulhaber noted that if p is odd, then

1^p + 2^p + 3^p + 4^p + … + n^p  

Is a polynomial function of

a = 1 + 2 + 3 + … + n = n(n+1)/2

In particular

1³ + 2³ + 3³ + … + n³ = a²;

1⁵ + 2⁵ + 3⁵ + … + n⁵ = (4a³ – a²)/3;

1⁷ + 2⁷ + 3⁷ + … + n⁷ = (6a⁴ – 4a³ + a²)/3;

1⁹ + 2⁹ + 3⁹ + … + n⁹ = (16a⁵ – 20a⁴ + 12a³ - 3a²)/5;

1¹¹ + 2¹¹ + 3¹¹ + … n¹¹ = (16a⁶ – 32a⁵ + 34a⁴ – 20a³ + 5a2)/3;

If you ask me, these formulas are remarkable and stagger my mind to imagine how Faulhaber would have gone about deriving these formulas.

What is even more interesting is that Johann Faulhaber was a weaver by training in the German city of Ulm which has no general significance except for one important fact; it is the birthplace of perhaps the greatest imaginative mind of the twentieth century Albert Einstein.

Johann Faulhaber later took to surveying the city.

His greatest contribution to mathematics was this – calculating the sums of powers of integers which finds mention in the book of Jacob Bernoulli Ars Conjectandi.

Yet even he could not derive the general formula for sums of all powers of integers restraining himself only to the odd ones if you care to go back and review the formulas atop.

In his 1713 book Ars Conjectandi on page 97 under the heading of “Summae Potestatum” Jacob Bernoulli published an expression of the sum of the p powers of the first n integers.

They were expressed as (p+1)th-degree polynomial function of n.

He used the long letter S for “summa” (sum).

∫n = 1/nn + 1/2n

∫nn = 1/3n³ + ½nn + 1/6n

^∫n³ = 1/4n⁴ + 1/2n³ + 1/4nn

^∫n⁴ = 1/5n⁵ + 1/2n⁴ + 1/3n³ – 1/30n

The formulae go on in this manner till summa 10.

Whenever my bedtime stories have mathematics in them I urge the readers to make a leap for the blog page of Pan narrans as Whatsapp has no infrastructure to support mathematical equations.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg