A sum is the result of an Addition. For example, adding 1, 2, 3, and 4 gives the sum 10, written

(1) |

(2) |

A simple graphical proof of the sum can also be given. Construct a sequence of stacks of boxes, each 1 unit across and units high, where , 2, ..., . Now add a rotated copy on top, as in the above figure. Note that the resulting figure has Width and Height , and so has Area . The desired sum is half this, so the Area of the boxes in the sum is . Since the boxes are of unit width, this is also the value of the sum.

The sum can also be computed using the first Euler-Maclaurin Integration Formula

(3) |

(4) |

The general finite sum of integral Powers can be given by the expression

(5) |

An analytic solution for a sum of Powers of integers is

(6) |

(7) |

(8) |

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) |

Factoring the above equations results in

(19) | |

(20) | |

(21) | |

(22) | |

(23) | |

(24) | |

(25) | |

(26) | |

(27) | |

(28) |

(29) |

Sums of the following type can also be done analytically.

(30) | |||

(31) | |||

(32) |

By Induction, the sum for an arbitrary Power is

(33) |

Other analytic sums include

(34) |

(35) |

(36) |

so

(37) |

(38) |

(39) |

(40) |

(41) |

To minimize the sum of a set of squares of numbers about a given number

(42) |

(43) |

(44) |

**References**

Boyer, C. B. ``Pascal's Formula for the Sums of the Powers of the Integers.'' *Scripta Math.* **9**, 237-244, 1943.

Courant, R. and Robbins, H. ``The Sum of the First Squares.'' §1.4 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 14-15, 1996.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. *A=B.* Wellesley, MA: A. K. Peters, 1996.

© 1996-9

1999-05-26