Recently in maths chat a couple of discussions on integrals and infinite series lead to the following observations that might be related to function behavior and closed forms:
Case 1
Consider the following general convergent infinite series:
$$\sum_{k=1}^{\infty}f(k)$$
Since the series is convergent, the forward/backward/central finite differences always exists for consecutive terms. The forward finite difference of step size 1 is defined as $\Delta_1[g](x)=g(x+1)-g(x)$. Now let $\Delta_1 S(k)= f(k)$. Then by summing up the terms one by one (which basically proves the 2nd fundamental theorem of calculus of finite difference, since all intermediate terms will cancel out), we get
$$\lim_{R\to \infty}S(R+1) -S(1)=\sum_{k=1}^{\infty}f(k)$$
where because the series is convergent, $\lim_{R\to \infty}\Delta_1S(R)=0$ hence
$$\lim_{R\to \infty}S(R) -S(1)=\sum_{k=1}^{\infty}f(k)$$
Case 2
Consider the following generic (Riemannian) integral over an interval $(a,b)$. Using the 2nd fundamental theorem of calculus, we can find
$$\int_a^b f(x)dx=F(b)-F(a)$$
moreover, the theorem guarentee uniqueness of $F$ up to a constant if it exists (which that question for the elementary case is addressed by differential Galois theory)
Now since an infinite sum is basically a Lebesgue integral under the countable measure, and the structure of a solvable Galois group of a given polynomial will give you the exact details to find the roots, and differential Galois theory is a generalisation of Galois theory to differential equations. Therefore:
Since infinite sums and finite differences obey a version of the fundamental theorem of calculus, is $S$ guarenteed to be unique (up to some constant factor) and will treating it as a Lebesgue integral and applying differential Galois theory will thus allow us to compute the indefinite sums $$\sum_{x}f(x)$$ hence the closed form of not just the infinite series, but the whole family of sums with a given summand $f$?
Is it possible to generalise differential galois theory so that special functions can be included into the framework as if they are treated as new elementary functions in order to make more integrals computable algorithmetically?