https://github.com/tiagocavalcante/k-power-n-sum

Finds the formula for the sum of the kth powers of the first n natural numbers for any positive integer k
https://github.com/tiagocavalcante/k-power-n-sum

linear-algebra mathematics

Last synced: 12 days ago
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Finds the formula for the sum of the kth powers of the first n natural numbers for any positive integer k

• Host: GitHub
• URL: https://github.com/tiagocavalcante/k-power-n-sum
• Owner: TiagoCavalcante
• License: mit
• Created: 2022-10-20T23:41:21.000Z (over 2 years ago)
• Default Branch: main
• Last Pushed: 2022-10-21T14:40:46.000Z (over 2 years ago)
• Last Synced: 2024-12-01T13:48:49.087Z (2 months ago)
• Topics: linear-algebra, mathematics
• Language: Python
• Homepage:
• Size: 3.91 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • Funding: .github/FUNDING.yml
  • License: LICENSE

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README

        
# k-power-n-sum

Finds the formula for the sum of the kth powers of the first n natural numbers for any positive integer k

## How it works?

This generates the augmented matrix for the linear system of the formula assuming that the formula of the sum of k-th powers is a polynomial of degree k + 1.

## How fast is it?

For small values of k this is very fast, but SymPy gets really slow with big values of k, if you care about performance I'd solve the linear system with C++, using the ideas from [this article](https://dspace.cvut.cz/bitstream/handle/10467/95144/F8-BP-2021-Eyvazov-Emil-Emil_Eyvazov_Thesis.pdf).

Just an idea abot the performance:

```sh

$ time python3 main.py 

1^10 + ... + n^10 = n**11/11 + n**10/2 + 5*n**9/6 - n**7 + n**5 - n**3/2 + 5*n/66

real    0m0.403s

user    0m0.351s

sys     0m0.052s

$ time python3 main.py 

1^200 + ... + n^200 = n**201/201 + n**200/2 + 50*n**199/3 - ...

real    0m24.700s

user    0m24.545s

sys     0m0.154s

```