  In 2014, Ibrahim M Alabdulmohsin wrote a paper called "Summability Calculus"
where he developed a method to generalize sigma notation to non-integer upper
bounds. His paper included a theorem, known as Theorem 6.1.1 (denoted here as
Lemma 2.1 because of its simplicity and location in this paper), but doesn't
study it much. Another paper by Mueller and Schleicher also analyzed this
formula, but doesn't integrate or differentiate the formula and states some
specific applications. This paper will analyze the simple formula that
generalizes sigma notation to non-integer upper and lower bounds. We state and
prove this formula in Section 2. Section 3 states a few algebraic properties
for the sum and product formulae and shows how differentiation of sums and
products works. Because integrating a product is challenging, we only analyze
the integration of sums in the fourth part of Section 3. In Section 4 we apply
the formula in the second section to create analytic continuations for
functions defined as partial sums, formulate an infinite series representation
to any limit, create a great approximation for functions that approach a
certain limit, make an analytic continuation for products, and calculate the
sum of anti-derivatives. We then conclude with a discussion of the material of
this paper.
