11284

Using Bernoulli's Formula to Sum Powers of the Integers from 1 to n

The Gauss schoolboy problem is to add the numbers 1 to 100. The trick is to pair the numbers to get .
Now the next natural question is to find the sum of the squares of the numbers 1 to 100. Is there another trick? Yes, there is, and there is a famous rant about it:
"From this it will become clear how useless was the work of Ismaël Bullialdus spent on the compilation of his voluminous Arithmetica Infinitorum in which he did nothing more than compute with immense labor the sums of the first six powers, which is only a part of what we have accomplished in the space of a single page." Jakob Bernoulli
The method from that single page is now known as Bernoulli's formula; it gives the sum of the powers of the numbers from 1 to ,
.
For the sum of squares problem, the range is 100 and the power is 2. The first three Bernoulli numbers are . The total is .
In 1842, Ada Lovelace proposed an algorithm for calculating Bernoulli numbers [1] that can be regarded as the first computer program ever written.

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Reference
[1] S. Wolfram, "Untangling the Tale of Ada Lovelace," from Stephen Wolfram Blog—A Wolfram Web Resource. (Aug 29, 2016) blog.stephenwolfram.com/2015/12/untangling-the-tale-of-ada-lovelace.
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