Sum of a serie : 1/n^2 ... and others...

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Sum of a serie : 1/n^2 ... and others...

Postby Franck » 07 May 2015 13:43

The sum of a serie is the sum of alls terms of a sequence.
Of course, all terms can't be computed, so we have to limit the numbers of terms to add.

To add the first n terms of S = sum(1/(k^2)), we can write :

Code: Select all
: sumZ(n) { 0 n seq apply(#[ sq inv + ]) }

- 0 is pushed on the stack. This will be used to accumulate the sum.
- n seq creates (and pushes) a sequence of integers from 1 to n. A sequence is a collection
- #apply will apply something on each element of a collection ie push an element on the stack, then push "something" then call perform.
- Here, we apply a block, an object that implement an anonymous function. When a block is performed, code into the block is performed.
- So, for each element of the sequence, we perform "sq inv +", which accumulates the sum we are computing.

>sumZ(1000) .s
[1] (Float) 1.64393456668156

This approximates series sum value when all terms are added, which is : Pi^2 / 6

>Pi sq 6 / .s
[1] (Float) 1.64493406684823
[2] (Float) 1.64393456668156

But it would be cool to have a function that calculates the sum of a serie that is sent as a parameter.
Blocks can do this because if a block uses a parameter, it becomes a closure :

Code: Select all
: sumSerie(s, n) { 0 n seq apply(#[ s perform + ]) }

Here, the serie is sent as parameter and the block uses s.

Now, if we want to calculate our serie (1/(k^2)), we can send it as a parameter :
>sumSerie(#[ sq inv ], 1000) .s
[1] (Float) 1.64393456668156

If we want to appromimate the sum of serie T = sum(1/(k^4)), which value is Pi^4 / 90,
>sumSerie(#[ sq sq inv ], 1000) .s
[1] (Float) 1.08232323337831

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