.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/combinatorics/sequences/plot_02_bernoulli_power_sums.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_combinatorics_sequences_plot_02_bernoulli_power_sums.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_combinatorics_sequences_plot_02_bernoulli_power_sums.py:


Jacob Bernoulli's numbers and sums of powers
==================================================

Computes the Bernoulli numbers exactly, uses them to evaluate sums of
p-th powers in closed form, and repeats Bernoulli's own boast: the sum
of the tenth powers of the first 1000 integers.

.. GENERATED FROM PYTHON SOURCE LINES 11-13

.. code-block:: Python

    from mathematicskit.combinatorics import bernoulli_numbers, sum_of_powers








.. GENERATED FROM PYTHON SOURCE LINES 14-16

The first Bernoulli numbers
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 16-20

.. code-block:: Python


    for m, b in enumerate(bernoulli_numbers(12)):
        print(f"B_{m:<2d} = {b}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    B_0  = 1
    B_1  = 1/2
    B_2  = 1/6
    B_3  = 0
    B_4  = -1/30
    B_5  = 0
    B_6  = 1/42
    B_7  = 0
    B_8  = -1/30
    B_9  = 0
    B_10 = 5/66
    B_11 = 0
    B_12 = -691/2730




.. GENERATED FROM PYTHON SOURCE LINES 21-23

Closed-form power sums
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-30

.. code-block:: Python


    for p in range(5):
        formula = sum_of_powers(100, p)
        direct = sum(k**p for k in range(1, 101))
        print(f"sum of k^{p} for k = 1..100: formula {formula}, direct {direct}")

    print(f"\n1^10 + ... + 1000^10 = {sum_of_powers(1000, 10):,}")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    sum of k^0 for k = 1..100: formula 100, direct 100
    sum of k^1 for k = 1..100: formula 5050, direct 5050
    sum of k^2 for k = 1..100: formula 338350, direct 338350
    sum of k^3 for k = 1..100: formula 25502500, direct 25502500
    sum of k^4 for k = 1..100: formula 2050333330, direct 2050333330

    1^10 + ... + 1000^10 = 91,409,924,241,424,243,424,241,924,242,500





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_combinatorics_sequences_plot_02_bernoulli_power_sums.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_bernoulli_power_sums.ipynb <plot_02_bernoulli_power_sums.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_bernoulli_power_sums.py <plot_02_bernoulli_power_sums.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_bernoulli_power_sums.zip <plot_02_bernoulli_power_sums.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_