Converting a summation function into a continous function.

Question

Consider functions like this: $$\sum_{n=1}^xn=\frac{x\left(x+1\right)}{2}$$ For any polynomial the summation can be turned into a continuous one. https://en.wikipedia.org/wiki/Faulhaber%27s_formula
Also consider: $$\sum_{n=0}^x\sin n=\frac{\sin\left(\frac{x}{2}\right)\sin\left(\frac{\left(x+1\right)}{2}\right)}{\sin\left(\frac{1}{2}\right)}$$ Is there a general method in converting summations into continuous functions. If not general then sub categories like periodic functions, ect.

Answer 1

As explained here, one can define fractional summations by considering the limit of a sequence of polynomial approximations to the summand. While no closed form formula is possible when summing a general function, one can sometimes derive exact results of the sum to specific fractional numbers, e.g. minus -1/2. For example:

$$\sum_{n=1}^{-\frac{1}{2}}\frac{1}{n} = -2\log(2)$$

Fractional summation can be used to compute integrals, see e.g. here for a simple example.

Answer 2

Let $f(n)$ be the sum up to $n$ terms. On the graph of $f$, we can draw a straight line segment connecting $f(n)$ and $f(n+1)$. The function that plots this graph consists of piecewise connected linear functions and thus is continuous. More explicitly, if $n\leq x< n+1$, we define $$g(x) = [f(n+1)-f(n)]x + f(n) - [f(n+1)-f(n)]$$ This was derived just by finding the slope-intercept form of the line segment.