The integral sign began as Leibniz's stretched-out S for sum
In a 1675 letter, Gottfried Wilhelm Leibniz proposed a tall, elongated S to mark a sum of differentials, from the Latin for summing calculus. That symbol later became the integral sign. The plain summation sign we use today, a big Greek capital sigma, shows up eighty years later in the work of Leonhard Euler.
Summation simply means adding up a list of values, called summands or addends, to get a total. The list 1, 2, 4, 2 adds to 9, and because addition does not care about order or grouping, no brackets are needed. Numbers are not the only things that can be summed: functions, vectors, matrices and polynomials all qualify, as does any mathematical object with a sensible notion of plus. Adding infinitely many terms is a different matter, called a series, and depends on the idea of a limit.
Long sums can be written with an ellipsis, as in 1 + 2 + 3 up to 100, but sigma notation is more compact. An index variable starts at a lower bound written beneath the sigma, rises by one each step and stops at the upper bound written above. The letter chosen for the index is arbitrary, and when context makes things obvious the bounds are often dropped. Sums can also run over every value meeting some condition, and two sigmas stacked together express a double sum. A capital pi does the same job for multiplication.
Edge cases have tidy conventions. A sum with a single term equals that term. A sum with no terms at all, the empty sum, equals zero, because zero is the value that leaves any addition unchanged. An algebraic sum mixes positive and negative terms, adding some and subtracting others. A recurring goal is finding a closed formula that replaces a long sum, though such formulas do not always exist.
The history of the notation is patchy. After Leibniz, the renaming of his symbol to integral grew out of exchanges with Johann Bernoulli. Euler used sigma in 1755, Lagrange in 1772, and Fourier and Jacobi in 1829, with Fourier adding explicit upper and lower limits. A capital S was also widely used for series around 1823.
Source: Summation
