Is it always true that no closed forms exists for any divergent series?

Question

Having seen many questions regarding finding closed form of integrals or infinite series, and some users providing either the final answer or detailed solution, and also reading how one finds a closed form of an equation

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

Some users are mind bogglingly skilled at integration. How did they get there?

After the MSE suggestions popped up when typing the question, it seems this question is an extension of this one Is there an algorithm to determine if a closed form solution exists? which was known to be there are no known ways to detect the existence of close forms for any general mathematical expression

(maybe a stupid subquestion, but I have seen many weird exceptional things in maths thus starting to wonder)

is it always true that no closed forms exists for any divergent series?

Answer 1

The divergent series

$$\sum_{i=1}^n 1$$

has the closed form

$$n$$

That was so easy I doubt that was your actual question.

Answer 2

Let us consider two types of divergent series.

1. Divergent series that can be summed up by somehow averaging over partial sums (Cesaro, Abel summation techniques)

2. Divergent series that diverge to positive or negative infinity.

Type 1

Definitely, one can assign a value to such divergent series, taking Cesaro or Abel average of the partial sums. For instance, $\sum_{k=1}^\infty (-1)^k$ can be assigned the value of $-1/2$ this way.

Type 2

Even if a series diverge to infinity, it can be assigned a "sum" using more complicated techniques. This is called "regularization". For instance, this way the sum $\sum_{k=1}^\infty 1$ can be also assigned the value $-1/2$.

But one should not forget that in the later case there is also an infinite term that we threw away. By assigning certain divergent series or integrals specific symbols, one can write many of them as "closed form" as well.